Don't know what I asked Google earlier, but here's what it says right now. (FWLIW, seems to attribute it to Reddit sources.)

Privately owned MLB stadiums are rare, as most are publicly owned or heavily subsidized. Notable examples of privately owned or financed stadiums include Oracle Park (San Francisco Giants), Dodger Stadium (Los Angeles Dodgers), and Wrigley Field (Chicago Cubs). Oracle Park was the first privately financed ballpark since Dodger Stadium.

Oracle Park (San Francisco Giants): Privately financed and opened in 2000.
Dodger Stadium (Los Angeles Dodgers): Built privately by Walter O'Malley in 1962.
Wrigley Field (Chicago Cubs): Privately owned by the team.
Future A's Ballpark (Las Vegas): Planned as a privately financed project.

While some teams own their stadiums, many still benefit from public infrastructure improvements, making the definition of "private" sometimes complex.