"Share me" or "Share with me"?

"share me" indicates that it's you who gets shared. Usually that's not something you see outside of slavery and sexual relations ;)
So yes, you're quite correct that it's incorrect to use the term in the context you show.

But as with so many things, it's becoming ever more common in the day and age where cellphone text messages and twitter messages are rewriting the rules of English grammar (and spelling).

Contra all the other answers: this is a normal, though unusual, example of a standard construction in English, where a bitransitive verb (one with a direct and an indirect object) has two different realisations:

Give the book to me <=> give me the book

Show the picture to her <=> show her the picture

Make a cup of tea for me <=> make me a cup of tea

The examples show that the indirect object can be with to or for. Examples with with are rare, which is why I described this as unusual.

As you said, share with you something OR share something with you

constructs are correct. The latter being more colloquial.

sharing you is to distribute(or well, share) you with something/one