I am the biggest believer in the DM having the absolute right to say "No" to the players that you are ever likely to run into. That said, I believe that all rights come with the inherent responsibility to use those rights as responsibly as you can. So, I do believe that I can say to my players, "No, you can't do that" at the table, but I also believe that I have to have a compelling reason to do so.
One person consistently damaging the game for the rest of the players would be a strong enough reason.
Players don't generally use notes at my table, but it has happened. I have used notes for major exposition. When the fighter (whose background was that he was an amnesiac) recovered his memory, I had a sheet prepared for him. Then I let the player deal with how he wanted to handle the information. It was fun for everyone, and resulted in a flurry of more carefully devised backgrounds.
I had one game back in 2e where one player was a rogue who pretended to be a ranger. It wasn't to go against the party; it was simply the character concept. And it was fun. He didn't rely on note-passing though. He simply said what he wanted to do.
Player 1: "I try to climb the wall."
Player 2: "But you're a ranger! Rangers can't climb walls!"
DM: "There's some effort involved, but you manage to scramble up."
or
Player 1: "Do I notice any tracks?"
DM: "No"
The strange thing was that, no matter how often the "ranger" disabled a trap or climbed a wall, and no matter how bad he was at tracking, the other players didn't catch on until the rogue's alignment changed (due to a
Helm of Opposite Alignment) and he felt compelled to tell them the truth.
Waaayyyy back in 1e, I once participated in a game where there was a lot of note-passing in the first session. The players didn't know what sort of characters each had made, and they met in the Standard Adventurer's Inn. Of course, they players were only allowed to communicate with the DM via notes, while the DM described what happened, and who said what. The upshot was that the party that resulted included some NPCs....simply because it wasn't always possible to tell who was a PC and who was not. After the party was formed, the game became a "normal" game -- the players simply announced their actions. But it was a heck of a lot of fun, and resulted in PCs fighting PCs in the brawl before everything was sorted.
So, notes can be fun if used sparingly, or for a specific effect. And, again, I hesitate to tell my players that they cannot do something unless I have a very good reason.
YMMV. YDMB.