Appraisers do comparisons of recent sales of similar homes in the area. That's how they arrive at an appraised value. That's the number that matters to the lender.
I was a real estate agent for a few years.
Selling privately involves a few things.
1. Determining an asking price isn't that hard. An agent would typically do what an appraiser does and find comparable properties sold within a particular radius, often by school district, within the past 6 months. So if it's a two story, find a similar two story. Condition is a factor. The asking price is little more than an educated opinion of someone who understands the market. A formal appraisal costs money. A reputable realtor might give you an opinion on what he or she would price a house for listing. Ask two or three. We did it all the time. I lost many listings to agents who would list a price higher than I thought it would sell for, but that's the nature of the business.
2. The buyer is going to have an appraisal done anyway at his or her expense if they are financing. That's why I wouldn't spend the money.
3. If an agent is representing a buyer of a for sale by owner home, you might what to consider that the agent representing the buyer may ask for a commission. Here, it was usually 2.5%. If you as the seller say you aren't willing to pay, then the buyer will have to pay an additional amount to their agent or else their agent won't represent them. A good agent will have had them sign a buyer agency agreement which has a beginning and ending date, so you can't go behind an agent's back and make the deal. The buyer by contract will owe the agent the commission. It's part of the business.
4. Where these deals can get really tricky is during the inspection phase because if there are problems and the buyer isn't willing to buy in as is condition and the seller isn't willing to fix the problems, the deal is off. These are things covered in all real estate contracts. And it's how real estate agents really earn their money.
I've bought two properties using just a lawyer, but they were from my kids, and there wasn't going to be any negotiation. I knew what I was paying and what I was getting. It was easy, but that was family. Your situation it different.