This depends on the car. I'm also capable of doing an inspection myself. However, generally, no. I don't take the car to a mechanic. I look at a few items, check the history, get a feeling for the car by driving it and testing stuff, and talk to the person owning the car. Most major issues will be obvious from simple testing.
In the end, a used car is a lottery. Also, paying to have a car inspected can be costly and the person selling the car might not give a shit about selling it to you if they find any problems.
Let me give you a likely scenario:
Person is selling a used car. You go to them and look it over. Drive it and it all seems relatively okay. You ask to have it inspected and will pay the fee. The owner says okay. This is where the hypothetical splits:
1) The mechanic finds issues. Maybe they're major, maybe they're minor. You want to lower the price of the car because of that. The owner refuses and you refuse to buy the car for the price they're asking. If you hadn't done the inspection, you might have bought the car.
2) The mechanic finds no issues. Now you have to buy the car for the price they were asking. In fact, the owner may even ask more now because they just found out their car is in great shape from a mechanic!
I don't really pay to do inspections beyond myself. It's generally not worth the cost and anything that I find I'll handle myself. :/ A $100-150 inspection per car is a lot of dough. I can fix some issues with that money. :/
How many cars am I going to pay to have inspections done on? The cars I like to buy are very in demand cars with the seller frequently having no issue selling them unless they're totally POS. (In which case, you can spot them from a mile away)
Again, it really depends on the price and type of cars but for the cars I am frequently looking at... nah.