Do you have to host a game to count as beating it?

I mainly play Dungeon Defenders by just joining someone else's game. Does beating levels like that count or do you have to host them from your tavern? I ask mainly because I want to unlock skins and stuff, but if it doesn't count for achievements or something, that'd be nice to know as well.

Yes, joining another's game still counts toward achievements and map completions. Just make sure you are in Ranked mode (not Open mode), since achievements are only on Ranked.

When beating the Summit, everyone gets credit of the ending for the person who lands the killing blow on the Dragon (not the character you're playing as). This only affects the achievement where you must see all 4 hero endings. Other than this tiny caveat, it does not matter who hosts the game - everyone gets equal credit.