SQL Sampling: one element from each bucket

Here is a simulation of the basic setup i have: each person can hold multiple possessions.
Persons table:

id  name
1   Carl
2   Sam
3   Tom
4   Jack

Possessions table:

possession personId
car        2
shoes      2
shovel     2
tent       3
matches    3
axe        4

I want to generate a random set of possessions belonging to a random set of people, one possession per person.

So, in a non-SQL world I would generate a set of N random people and then pick a random possession for each person in the set. But how do I implement that in SQL semantics?

I thought of getting a random sample of possessions with some variation of:

SELECT * FROM Posessions WHERE 0.01 >= RAND()

And then filtering out duplicate persons, but that is no good as it will favor persons with large number of possessions in the end, and I want each person to have equal chance of being selected.

Is there a canonical way to solve this?

P.S. Person contains ~50000 entities and Possession contains ~2500000 entities, but i only need to perform this sampling once, so it can be somewhat slow.