In most of West Africa you can find buses and bush taxis (taxi-brousse) to take you between cities. SMTB buses run from Ouaga to Diapaga several days a week. But there is no public transportaion in the parks. If you have an international driving license you can rent a car in Ouagadougou or Niamey. Otherwise you will need to rent a car and a driver. You can do this in the city and have him drive you to the park, or ask your guide to arrange a car and driver.
Some guides have their own vehicles. Otherwise your guide can arrange a driver, whom you will pay separately. An air-conditioned 4x4 with driver runs 50,000 francs ($100) per day, non-air conditioned 40,000 ($80). You will also pay for the fuel. You may want both a driver and a guide, although if your party has more than two people you will be cramped in the car. The driver watches the road; the guide watches for animals. On bad roads like these it takes skill to do both at once! In any case, have someone else drive. If you drive yourself, you will miss seeing a lot.
You really don't want to tackle this sort of road yourself!
Expect any car to look old and beat-up. Most cars in West Africa are very-well-used diesel 4x4s, often referred to as "Adieu la France" because they said good-bye to France instead of dying gently in a junkyard. Some are quite dependable; some are not, and you may be asked to cough up the cash to replace the drive shaft!
Your guide will probably sit in front next to the driver so he can watch the road and spot animals for you. This means you will be in the back seat, which is not bad for two people, but gets cozy for three. Be sure you can hear the guide and see where he is pointing, especially if your French is less than perfect. If you are prone to motion sickness, the dirt roads will certainly exacerbate it. Some cars are air-conditioned; some are not. If the dust clogs the air-conditioning system, and you open the windows for air, expect to be orange by the time you reach your destination.
Make sure you have plenty of water in the car with you. Even though you aren't exercising on the road, you can still get dehydrated very easily.
Despite all these caveats, a private car is the only feasible way to get into the parks, and in retrospect you will say it was worth it!