Most often, yes. And in some cases, the inspections are optional, depending on whether or not you're financing the purchase, and what kind of loan, or paying cash, etc.
VA buyers are NOT ALLOWED to pay for pest inspections. It is the Seller's responsibility to pay for the inspection and provide a Pest Clearance as well.
Most inspectors, at least the ones I use, will not provide a different report whether they're representing the buyers or the sellers, or who actually requested the inspections.
There are inspectors, and there are GOOD inspectors. It's good to know which ones are which, especially if you're the Buyer.
I hope that helps.