I'm not sure I have a general, abstract answer. But here's a few case studies:
https://en.wikipedia.org/wiki/Impact_of_the_privatisation_of...
About the last: the (partial) privatisation of British Rail was a mess. Far from a textbook case. But still ridership numbers that had been in seeming terminal decline picked up in absolute terms, and rail's relative share of the overall transportation market increases total.
https://www.econlib.org/library/Enc1/AirlineDeregulation.htm...
The impact of deregulation of (American) airlines has been tremendous. Flying is now cheaper than ever. But to address my point: flying is still very strictly regulated, just less so than before (especially in the areas of pricing and competition).
https://www.econlib.org/library/Enc/TruckingDeregulation.htm...
https://www.econlib.org/library/Enc/SurfaceFreightTransporta...
Despite common complaints on the Internet, the US is actually still really, really good at running railroads. It's just that their area of competence is freight rail. Passenger rail in the US is anemic.
Guess which part is in government hands, and which one is comparatively free market and less regulated? However, neither area is completely free market or completely state controlled.
Another example: a few years ago Germany legalized long distance busses. The old ban was a hangover from the Nazi era.
https://www.dw.com/en/regulations-eased-on-long-distance-bus... is an overview article from just before the legalization of busses came into effect. I leave it as an exercise for the reader to find some sources from afterwards. Google Translate might be helpful, if you don't read German.
And, of course, legalization and deregulation here just mean different (and a bit less) regulation. There's still plenty of rules. We are talking about Germany, after all.