What type of paint is best, or what paint technique?

Could you stain the piece brown first? And then take a cream white or pure white color and dry brush over the board once it’s stained. You’ll get that whitewashed look. If you look at the FB page for Choxie’s craft ‘n crate in Georgia, you’ll see some welcome signs that have this white washed look to them. Very easy to DIY paint on your own.

DecoArt makes a great whitewash-style paint. It's in their "Vintage Effects" line :)

I would mix some gray into whitewash as dried driftwood is grayish. Whitewash is traditionally a mix of 50% water and 50% latex paint. Give the wood a quick sand to open up the wood fibers, then use a large brush to apply and rags to wipe off. If you want more wash on the wood, repeat and let sit a bit longer, then wipe off.

If you want some areas white and others grayed look, You could use a container to keep some of the paint white and apply the 2 colors to the pieces.

If you want it really grayed, I would get some Minwax gray stain, and then do a bit of whitewash over it.

I would use a flat white paint unless you want a bit of sheen then satin maybe. An old roughed up brush, a piece of cardboard and some paper towels are all you need. Go slow and sparingly, dip the tips of the brush, wipe most off on the cardboard and dry brush the wood. What you want to avoid here is a starting splotch of paint. (Been there) and you can't fix it. Just keep layering on the paint until you are happy with the look. A tip I use is every so often walk away from it and look at it from a bit of distance, that can make such a difference to what is enough. Good luck, sounds like fun. :)

I've seen others water down the white pain and rub it on with a rag instead of paint brush. Better control of the look you want. Try the back side first.