You can set it up to work either way. By default, I think, an asylum is set so that only a moderator can add new tags, but any member can apply existing tags to entries.
So if you tagged this entry "tales of the abyss" and "dessert" and then Mith posted her Vossler cheesecake recipe, she'd be able to add the dessert tag to it but would still need you to add the "final fantasy xii" tag.
If that sounds like too much trouble, you should be able to change the settings so that anyone can add tags any way they like. I'm not entirely sure where the setting is for that, but it's probably in the FAQ. ^^