How long they take to sink depends on several factors. Species of the wood, how dry it is, how thick it is. Can take weeks, can take a day or two. You'll find out. A piece of the wood they often sell for terrariums took months and months to sink, and many water changes, as I was not sure it had not been treated with something. Looks nice now, but I would not get this kind of wood again.
I had a nice piece of grapevine that took two months before it would stay below the surface by itself, and longer still before it would stay on the bottom by itself. I had worked it over with a dremel tool, to sculpt it a bit, and some of the heartwood rotted out, but the remainder has been in a tank for a year and so far, mostly intact. A small piece broke off, but the rest is good.
If your woods are hard woods, most of those sink fairly fast. If not, it will take longer. You can put a bit of bleach in the bucket and do without an air stone if you wish. Bleach stops the water going nasty. It will also remove colour a bit. With beach wood that should not matter much, it's already pretty bleached out. Rinse well and use extra dechlorinator if you bleach the wood, when you go to use it.