No, because they'll stick their Possession in an Archive and wait for you to pay off your own debt before playing it?
I have not done more than read the cards and the Debt entry in the wiki, but here is what I am thinking:
I build a reliable Possession deck (at least one Possession play per turn).
You gain the first province after I have at least one Possession.
I immediately bid 40 debt.
When I Possess your deck, I am making decisions that you officially carry out. As such, you have no debt to pay back.
I can keep
gaining cards after
you buy them.
You Possess my deck, you are making decisions that I officially carry out. I officially have 40 debt and those must be paid back before I can buy anything; you can gain no cards until the debt is paid.
I could be parsing wrongly here, but I think this is how you resolve debt & Possession.