Valencia Airport is compact by international standards, which works in your favour — there's really only one place a private transfer driver can wait, and it's the same place for every flight, every terminal exit. The uncertainty most travellers feel beforehand usually has more to do with not having flown into an unfamiliar airport than with the airport itself being complicated.
Where Arrivals Actually Puts You
After collecting luggage and clearing customs, every arriving passenger exits through the same set of doors into a single public Arrivals hall. There's no separate "private transfer" exit or hidden door — you'll walk out into the same open space as everyone else, which is exactly where your driver is waiting.