Can I assign tasks to someone?
Yes. Open the task and pick an owner from the people in your wedding: you, your partner, both, or a collaborator. You can filter the list by owner to see at a glance who has what.
Assigning a task does two things: it stops both of you assuming the other one is on it, and it lets you see the list filtered by person.
How to do it
Open the task and choose the owner. The options are the people already part of your wedding — you, your partner and any collaborators you've added — plus "Both" and "Unassigned".
Seeing only yours
In the task list you can filter by owner, or group the view by person. Tasks OPHIRA suggests start with no owner until someone takes them.