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I suspect the answer is "not in general, unless you're willing to pump extra money into the payment-extracting mechanism". Depending on how generally you define "free rider problem", at least some examples of the problem are likely to be captured by Holmström's theorem: a system which pays each member of a team to to provide inputs to produce output can't balance its budget (exactly split the output's value among the team members), be self-enforcing, and Pareto efficient.
I think your fresh paint example is susceptible to Holmströmean logic. Suppose you've found a painter who's willing to do an amount of painting proportional to the money they're given; you're willing to pay for $40 worth of painting but your two housemates would only pay for $20 worth of painting. If you propose to either housemate that you pay $20 to the painter while the housemates pay $10 each, the housemate will (assuming they act as homo economicus spherical cows) notice that if they don't pay anything, $30 of painting will still get done, more than the $20 they wanted in the first place. So they won't pay anything, as you expected.