You'll definitely get the most money selling it yourself. You don't need to sell it from home either. That way the buyer wouldn't know how to contact you for any reason.
I sold my wife's car in a Wal-Mart parking lot in less than 30 minutes with multiple interested buyers.
You just need the title to be able to sign it over to the buyer. In addition you just need a bill of sale form which you can get online anywhere. The bill should state selling the car "as is". The only identifier the buyer will have is your name on the title and bill of sale. No address, phone, number, etc.
If you decide to sell on your own, accept ONLY cash, no checks.
Ask for more than you'll accept so you can lower the price. For example, if you want to get $6,000, maybe you ask for $6,500 so you can drop the price when the buyer counteroffers.
And thanks for giving me the idea for my next article!