I co-own a PA-28 & checked this out a few months back.
In short:
If it's on a Public C of A, yes
If it's on a Private C of A, No
You can train on a Private C of A if you fully own the a/c, but not if you're a partial owner.
Once you have a lisence, you can renew it on a co-owned a/c but for some reason, you can't earn it in the first place.
I got this info straight from the horses mouth (CAA) & they confirmed it by pointing me to the relevent docs.
AIS form 4W049