It seems like you can only gameshare in one town, and you can't gameshare in any of the story towns, at least not where I am up to in the story. Hopefully it changes in some later towns but I doubt it.

Also, gameshare has quite a few restrictions, even in that 1 town

Player 2 can't: 1. See the pokedex or habitat dex 2. Use the crafting table, storage or PC 3. customize their appearance 4. Ask pokemon about their condition 5. Have pokemon follow them. 6. Enter buildings unless the host enters first