That is a great question. I'm currently dealing with a transaction with a home on a Private Way. There are 2 main items you need to know. First, is there a maintenance agreement between the city of Arlington & the residents of the private road. Second, how are you going to finance the transaction?
If you're financing via FHA you will need this agreement in hand. If you're buying via conventional or local bank it's a question to ask the Loan Officer or their underwriting department. Finally if you're buying cash you have no problem.
I hope this helped and if you have any further questions I'd be glad to assist.