Today I have opened gossip again, and Private chats loaded immediately, including previous ones. At this point I am unsure what exactly was broken and how can I help you to reproduce this issue, sorry.
Discussion
It might be due to changes I have made to "My relays" were not applied before application was restarted. Current (known to work) relay configuration looks like following:

The only significant change I have made that I remember is that I previously had "Read" disabled for DM relays.
For a long time I've gotten similar feedback from new gossip users that nothing shows up in their feeds. But later on it starts working. And I've never been able to reproduce when I start from scratch. So this is probably just another case of the same syndrome.