For the rosewood, you generally leave it and then after the rest of the finishing is done pop some lemon oil on it and buff it back, then re-oil when it starts to look dry as maintenance. You can also use fretboard wax over the top of it.
For the maple boards they typically get some truoil or what ever finish you use on top.
EDIT: My Fretboard started off like yours, but i couldn't deal with the florid inlay and the pearloid scratchplate, so I stripped the original board off, made my own from ebony and added the MOP basic fretmarkers. I scuffed the scratchplate then blacked it out to contrast the PE logo. It was a lot of work but the paint was worse!