Does each member of my team need a Cloze subscription to contribute to a Project?

Yes.

Each Project or Deal has a unique Team. In order to contribute your interactions or observe you must also join the Project (or Deal) Team with your own Cloze Pro or Cloze Business account. 

If some of your team members have their own Cloze Pro subscription they can use all of the Cloze Business features as a member of your team.