Can friends join your world in co-op?

The invite goes both ways. You can pull a friend into your instance, or drop into theirs and play in their world instead. Nothing about the world stops being yours while they are in it. It stays private, so the people wandering around your map are the ones you asked in.

What you do together once inside is ordinary play. Exploration routes get shared, and creatures get caught with both of you working the same encounter rather than queuing for turns. Farm work carries a direct incentive to visit someone else's world: till their soil or help out on their farm and you take a portion of that harvest home.

Sanctum dungeons need a second player

Some content will not open for a solo player at all. Certain Sanctum puzzle dungeons have a minimum party size of two, so an invite is the entry requirement rather than a convenience. The puzzles themselves are built for two players working with their Aniimo, which means bringing a friend in is less about speed and more about having the second set of hands the design assumes.

Bosses in co-op

Boss fights are open to a group. World bosses and Alpha Aniimo bosses can both be taken on cooperatively, so a friend in your instance is useful for the hardest encounters on your map and not just the roaming ones.

Not settled: How many friends can be in one world at the same time is not settled. The only party number attached to co-op is the two-player minimum on certain Sanctum dungeons.