Let's say the war ends "soon" Ukraine gets chopped up in whatever configuration - don't think it matters much for our hypothetical scenario.
Russia has an economy which were adapted for the sanctions and avoiding them, and tuned towards military production. I do not think it's entirely militarized economy as some westerners wants us to think. It will have a large military, and a large group of citizens with appetite for projecting the empire.
But nothing to do.
I'm sure the economy will get rewind some, but it won't be like before the war. I'm also sure one focus will be drones. For both civilian and military applications - there will be lots of drone pilots too.
Anyway. The army. Quite a few soldiers will move to special forces units and PMCs. I think they will be used on various regions of the globe, especially in Africa and the Middle East. As fighting force, as advisors (training and commanding roles), as drone operators, as trainers of local drone operators. Russia will greatly enhance her reach and power projection with them. Pair them with BRICS or other institutions, they might offer economic aid (negotiating for multilateral loans, same how the US does with IMF).
Have to interject for a moment. Russia as an empire does the same as US - builds and maintains a client empire, and she creates the tools for it, just as the US had to in the past 100 years. See the politics thread for the client empire thing.

To counter them the West will use the battle hardened and willing Ukrainians these operators after the war cynic shells, lost their ideological motivation, fueled only by hatred, and perhaps the money they get as they already did against Wagner in Africa.

Will we see development in private wars, initiated by private companies? Similar to cyberpunk wars. It's not that it doesn't have historical precedence, see * India Companies.