Hello Igor,
To refer your question, it is quite interesting. To me, I would like to vote the correct sentences as below:
- Mary and we are going to the store.
- Mary and they are going to the store.
To refer these sentences, in the beginning of sentences, both are used in a kind of subject form which is called "Compound Subject".
In my opinion, in general, most of people (including myself, but I mainly made mistakes when I was young) are not familiar with this kind of grammar constructure of subject form , e.g.: Mary and "we " or "they". Due to this situation, people are used to build up a sentence with using a subject (E.g.: Mary) / subject pronoun (e.g.:"we" or "they") plus an object pronoun (e.g.: "us" or "them") to be combined together when they need to figure out the sentences.
I hope it can help you!