I would guess a "private" launch to Mars would be directly or indirectly subsidized by Uncle Sugar, the way many "private" sports stadia are.

I don't have the slightest idea as to the situation where you are, but tax breaks, loan guarantees, and subsidies are common for those things, with ownership residing in private hands. In effect the public takes the chances.

What do you think the incentive for a private trip to Mars would be?

I think one of the ways it could be done more efficiently privately is that there doesn't have to be all the CYA stuff a public agency has to do. If you're a public bureaucracy official it makes sense to include everything you can to protect yourself from any conceivable criticism from politicians and the public. That's only prudent. Some of that might well be unnecessary for a private corporation.