I don't believe there is any equivalent here. People race planes, but military planes are forbidden I believe.
here's the kind of thing offered here : http://aircombat.com/about/our-aircraft/
fast and aerobatic planes, but not fighters.
There isn't any specific prohibition against owning US jets AFAIK, but not many ever hit the market. WWII era stuff is common. Some of the early jets (e.g. F-86, F-100s, F-5/T-38) exist in private hands.
In general, the Russian stuff translates better to private ownership because there is much less specialized support equipment or maintaince required - in general the soviet's built stuff for a real SHTF scenario, where-as the US was much more willing to complicate the design to gain a small edge.
MiG-29 recently made headlines in India, the first and one of the largest export user, when one crashed
http://www.thehindu.com/news/national/other-states/mig29-cra...