It sounds like the market socialists you’ve been talking to haven’t been socialists if they’re in favor of private property, that’s strictly a capitalist position. They’re probably just welfare capitalists.
An actual market socialist is against private entities owning the means of production, they’re owned communally by some mechanism (be it some democratically run cooperative, the state, etc .) It wouldn’t be a group of stakeholders that are a separate, private entity disconnected from the workers (though the state arguably is an entity like that, and that’s where the line between state socialism and state capitalism gets blurry).
“from a private third party” where? A (non-foolish) socialist would advocate for rules against renting people, just like we’re not allowed to buy people right now.
That would mean there would be no private third parties that are renting out factories of rented workers.
If what you’re saying is “from a private third party outside the socialist space”, then that’s a problem for all kinds of socialist spaces. We can’t control productive forces outside of the space we have domain over.