No no... I don't expect a chauffeur. When I was in India back in April/May, I hired drivers in multiple cities and it worked out great. I just wasn't sure if this was something normally done between BOM and Pune.
I've done this in the past. Note that the driver will have no clue how to navigate the other city. Take a driver from BOM to PNQ and he'll not be able to drive you to your destination without a lot of direction from you or people along the side of the road or the people you are visiting. With this in mind give yourself LOTS of extra time.
My last trip between the two was via air and that is the way I'd do it again. Much quicker taking everything into account.