Which one is more idiomatic: ‘valid concern’ or ‘legitimate concern’?

Solution 1:

The following Google 2grams would indicate that 'legitimate concern' is the more frequently used of your alternatives, but also that 'proper concern' where 'proper' reinforces that people should have concern here (rather than it's all right to have it) is also idiomatic.

The following senses of proper [Cambridge Dictionary] are relevant:

proper [adjective] ... [ before noun ] real, satisfactory, suitable, or correct: ...

• She likes everything to be in its proper place.

But especially:

proper [adjective] (socially acceptable) showing standards of behaviour that are socially and morally acceptable:

• In those days it was considered not quite proper for young ladies to be seen talking to men in public.

and from Macmillan:

proper ...

2 considered to be morally good

• I don’t think it would be proper for me to say any more.
• I’m sure you will all agree that this is the only proper course of action.
• right and proper: It’s only right and proper that his family should be present.

It is a formal and some would consider dated sense, but the Ngram and high listing in both the dictionary senses show that it is far from moribund.

Although legitimate does carry the sense of 'being good and worth considering' as well as 'lawful', as seen in this sense given by Merriam-Webster:

legitimate ...

b: genuinely good, impressive, or capable of success

• And if he can fix his start … he is a legitimate contender for gold at the Tokyo 2020 Olympics. — Sean Ingle

it does not carry the 'duty-bound', moral obligation sense that 'proper' does.

Solution 2:

While both valid and legitimate can be used in this context, the problem with them is that, in the present-day usage, they involve a certain amount of ambiguity. Sometimes, one may say that a concern is valid or legitimate when one in fact shares it, and thinks that everybody should. Sometimes, however, saying that a concern is valid or legitimate, if its meaning were fully explicated, would amount to something like this:

It is perfectly understandable that you have this concern. You should not feel that is silly or crazy to have this concern. You have every right to express this concern and to have other people listen attentively to what you have to say about the matter. I am, however, not saying anything at the moment as to whether everybody ought to have that same concern, or whether I, in particular, have it.

If one wants to convey the idea that there is something rationally compelling about the concern, and that everybody ought to share it, then well-justified concern is more likely to accomplish the purpose unambiguously. What would probably be even better is to not insist on an adjective, but instead say something like 'a concern for which there are good reasons'.