Do I have to repeat the same subject after "and" when introducing a new verb?

TLDR: You are basically correct, but because the things you are connecting with and are not the entire verb but only a part of it, it can feel ambiguous unless you repeat some pieces. You should also remember that while Spanish allows subjects to be omitted whenever no confusion would arise, English only allows that omission under extremely limited circumstances. This is one where English does allow that omission, though, provided that you’re careful enough.

Abstract Discussion

When you have a compound verb sharing the same subject, there is no need to repeat the same subject. This is the same in English as it is in Spanish. So both these kinds of thing are perfectly common in English:

• I called and left a message.
• Every morning, Maria gets up before her kids and makes them a fancy breakfast.

Besides having omitted a to in need to, the trouble you’re having with your own sentence is that you are using a negative, and so it isn’t always clear whether it’s best to use:

1. not A and B
2. not A nor B
3. not A and not B
4. neither A nor B

The first one is ambiguous because one does not know for sure which of these two applies:

4. (did not A) and (did B)
5. did neither (A nor B)

Leading to things like this:

• I didn’t forgive and forget.
• I didn’t forgive nor forget.
• I didn’t forgive and didn’t forget.
• I neither forgave nor forgot.
• Neither did I forgive nor forget.

Sometimes it feels clearer in English to repeat the subject for this situation:

• I did not forgive, and I did not forget.
• I did not forgive, nor did I forget.
• I did not forgive; neither did I forget.
• Neither did I forgive nor did I forget.

If you want only the first part to be negative and the second positive, using but for a conjunction works better:

7. not A but B

Thereby producing:

• I didn’t forget but did forgive.
• I didn’t forget but I did forgive.

You can also use yet for a conjunction here in English:

8. not A yet B

Leading to:

• I didn’t forget yet did forgive.
• I didn’t forget yet I did forgive.

Be aware, however, that this elevates the tone somewhat because yet is less commonly used than is but here.

Concrete Application

Your original is trickier because the first verb phrase is really need to forget and the second is must forget, both negated. You’re actually trying to join together two “helper” verbs (need to and must) with a shared target infinitive (forget) which both helper parts are expected to apply to equally. That means that the and does not apply to entire verbal phrase following, only to a piece of it. This leads to a slight possible ambiguity which is most easily clarified by repeating the subject.

Moreover, one is a regular tensed verb that takes a to-infinitive complement but the other is an untensed modal verb that takes a complement of a bare infinitive, which is another potential source of confusion.

This makes it hard to find the right place to put the and. You need to put it after the need to because the to shouldn’t really apply to the half of the compound verb with a modal.

• You don’t need to — and must not — forget.

This is smoother with a repeated subject, although both of these are somewhat formal or literary:

• You neither need to — nor must you — forget.
• You don’t need to forget, nor must you.

This is much easier if both pieces you are trying to join are the same sort of thing. For example, with two auxiliary verbs both taking simple infinitive complements:

• You can and must forget it.
• I could and did forget it.
• I shouldn’t and mustn’t forget it.
• I shall not and will not forget it.
• I shan’t and won’t forget it.
• I need not and must not forget it.
  [NB: When you negate need to this way, it becomes a true modal and so you may skip the to.]
• He need not and must not forget it.

By the way, your Spanish example would have worked a little better if you’d used some named third-person subject, since there’s no reason to use tú there in Spanish apart from emphasis, since it’s built into the verb conjugation already:

• No necesitas y no debes olvidar.
• Ni necesitas ni debes olvidar.

But with an explicitly named third-person subject, you then have a scenario that better maps to English usage:

• María no necesita y no debe olvidar.
• María ni necesita ni debe olvidar.

Because you are using need to in English and so have a verb plus a particle in one half but not in the other, a situation more parallel to that in Spanish would be if one were to use tienes que for the first part and only a simple debes in the second. I leave it to your own ear to decide whether these sound nice to you:

• María no tiene que y no debe olvidarlo.
• María ni tiene que ni debe olvidarlo.

Notice though how in the first sentence of my previous paragraph, I used a compound verb with a shared subject: You applies to both are using and have in that sentence, joined with an and. So you can and often do do these things in English.

It just becomes trickier when the pieces you are joining together are ambiguous because they are not the entire verbal phrase, only a piece of them. When there are ambiguities of precedence of operators like and in speech, we cannot use parentheses as we might in algebra. The best strategy in human language for resolving ambiguities caused by that sort of thing is to reword into a longer phrasing.

Postscript

Now you’ve gone and got the lyric of “La Cucaracha” running through my head:

      La cucaracha la cucaracha
         ya no puede caminar
      porque le falta y no tiene
          marijuana que fumar.

This one I blame on la María. :)

Neither is well written, because you are missing "to." They should be written:

You don't need to and mustn't forget.

You don't need to and you mustn't forget.

This is easier to see (or hear) if you separate out the "don't need" part on its own. "You don't need forget" (as opposed to "You don't need to forget") is wrong.

As for including the second "you," both are correct grammatically. It's a difference of emphasis and style.

Not having the second "you" makes the sentence more fluid, with the emphasis on not forgetting, and would probably be the default way to write and speak. Having the second "you" shifts the emphasis onto the two separate ideas of "you don't need" and "you must not."

What to repeat and what not to repeat -- this is not the best way to think about it. The basic rule is that you can conjoin things of the same category to form a constituent of that same category. Your example is complicated by the fact that it is not obvious just what the category of the conjoined constituents is. Also, it seems odd to use "and", so I'll change it to "but".

Consider this related example:

You don’t need to eat but you mustn’t forget to eat.

Look to the left and right of "but" to find the things conjoined. Here, they are two sentences -- the category of "You don't need to eat" is the same as the category of "You mustn't forget to eat", S, so the requirement for conjunction is satisfied.

Now consider:

You don’t need to eat but mustn’t forget to eat.

Look to the left and right of "but" to find the things conjoined. Here, they are two verb phrases, VP, and -- the category of "don't need to eat" is the same as the category of "mustn't forget to eat", VP, so the requirement for conjunction is satisfied.

So far so good. Now to come closer to the example you asked about, I have to do something about the "to eat", which I added in. In your example, this, or something comparable has been omitted from the VPs. The VPs are incomplete. To represent such incomplete categories, Gerald Gazdar suggested a notation that comes from the theory of Categorial Grammar, a slash, with the resultant category given to the left of the slash and the category of what is missing to the right of the slash. Here, I'll use VP/Inf to represent a verb phrase which is missing an infinitive.

Removing the infinitive from "need to eat" gives "need", of category VP/Inf. To become a full-fledged VP, this VP/Inf needs to be combined with something of category Inf. Thus "need" with category VP/Inf + "to eat" with category Inf yields "need to eat" with category VP.

Similarly, "mustn't forget" with category VP/Inf + "to eat" with category Inf yields "mustn't forget to eat" with category VP.

Now, we're ready to consider

You need but mustn't forget.

Look to the left and right of "but" to find the things conjoined. They are "need" with category VP/Inf and "mustn't forget" with category VP/Inf. Aha. The categories are the same, so this should be an acceptable conjunction. Furthermore, we know the category of the resultant phrase. "need but mustn't forget" must have the category "VP/Inf".

The position of the thing removed in such constructions is called a gap, and it is often marked by an intonation break, which could be represented by a comma in ordinary orthography:

You need [gap], but mustn't forget [gap],

Since a VP/Inf can be completed with an Inf, "to eat" for example, we expect also to get

You need [gap], but mustn't forget [gap], to eat.

and since a VP/Inf combined with an Inf is a VP, we know that this is going to be an S.

Your other example can be worked out in a similar way. This is a systemization of the Right Node Raising analysis originally proposed by Haj Ross. It was described in an article by Gerald Gazdar in Linguistic Inquiry.