Do I need to sand the new pressure-treated deck before staining it

I disagree with @Mike - I don't think sealing it will have much effect on the stain. Two reasons:

1. Deck stains are typically (I'm tempted to say always) pigment stains rather than dye stains. Those deposit pigment on the surface of the wood and penetration is minimal. A sealant shouldn't interfere with that.
2. A sealant will typically be a penetrating sealant not a film sealant, so it shouldn't interfere at all with the adherence of the later stain.

That said, I'm not sure you'll see much benefit from the sealant. Penetrating sealants will certainly give you a bit more time to clean up a spill but they won't prevent a stain: that's why tabletops are usually finished with a film finish, and if they're not, you use coasters and place mats. You wouldn't want to use a film finish to a deck because it will certainly peel in short order as the wood expands and contracts under rain and sunshine.

I wouldn't bother. If food gets spilled and you're worried, scrub right away with soap and water. I'll wager you won't notice any minor food marks after the stain has been applied.

You can likely answer your own question with a little experiment. Before committing the whole deck to large amounts of stain application try a small area on 1 or two boards where you've sanded on one side and the weathered grey is still on the other side. If you like the result you will know what to do. If you dislike the look or match then you can proceed to close up the stain can and go back to sanding and scrapping the rest of the deck surfaces. The place where you put the stain sample over the weathered grey can be sanded down to fresh wood right along with the rest of it.