Tried this today. Had high hopes but it turned out to be pretty uncomfortable for me, even with just one pit.
Personally I don't mind spitting out the pits at all, but for cases where it may be annoying (like in a fruit salad) I use a cherry pitter with a hopper: https://m.media-amazon.com/images/I/31vipTWimyL._AC_.jpg
Fun, too. Though it does tend to spray some cherry juice around.
When eating cherries, what do you do with the pits? You don't want to swallow them, but spitting them out one by one is inconvenient. The answer: use your upper lip.
Eat the cherry as normal, separating the pit with your teeth and tongue as usual, but once the pit is free use your tongue to tuck it up under your upper lip. Once it's there, it will stay safely out of the way while you chew and swallow the rest of the cherry. You can repeat this several times; depending on your mouth geometry you can probably hold six to ten pits.
Asking around and looking online, it seems a lot of people don't know you can do this? I think this is because the pits are not especially visible, and the only clue someone is doing this is that they're needing to remove pits much less often.
