The command that you need to hide chat from the UI is:
Keep in mind it's sv_cheats true protected.

Explanation:
The "hidehud" command is a bitmask. The help text for it is:

bitmask: 1=weapon selection, 2=flashlight, 4=all, 8=health, 16=player dead, 32=needssuit, 64=misc, 128=chat, 256=crosshair, 512=vehicle crosshair, 1024=in vehicle

So each number is hiding something from the user interface.
Yes, unfortunately, it doesn't look like you can hide players' messages individually, so if you want to hide players' chat messages in CS2 currently, you will have to sacrifice ALL chat messages, including who's throwing grenades.

Let's break down the help text a little bit in case if you need to hide something else:

1=weapon selection,

Not self-explanatory. Not only it hides weapon selection (weapons, grenades and their slots (dots), bomb, defuse kit presence), but also health, armor, kill amount, and amount of ammo. Also prevents slot1, slot2, slot3, etc commands from switching weapons.

2=flashlight,

CS2 doesn't have a flashlight icon. I looked if it changes anything else but it seems to be unused.

4=all,

Not self-explanatory. Doesn't hide all. Leaves money on-screen and keeps that annoying gradient at the bottom of the screen. Use cl_drawhud false instead of hidehud 4. Difference:

8=health,

CS2's health is hidden by weapon selection (1). I looked if it changes anything else but it seems to be unused.

16=player dead,

Hides ONLY weapon selection (weapons, grenades and their slots (dots), bomb, defuse kit presence) and amount of ammo. Also prevents slot1, slot2, slot3, etc commands from switching weapons.

32=needssuit,

CS2 doesn't have an H.E.V. suit like in Half-Life. I looked if it changes anything else but it seems to be unused.

64=misc,

Hides ONLY weapon selection (weapons, grenades and their slots (dots), bomb, defuse kit presence) but NOT amount of ammo. Doesn't prevent slot1, slot2, slot3, etc commands from switching weapons.

128=chat,

Hides all chat messages, icons of who is currently using voice chat and the "Voice Disabled" icon if you have voice chat disabled. Chat can still be opened using "y" and "u" keys by default ("messagemode" and "messagemode2" commands). Doesn't prevent the click sound when new chat messages appear.

256=crosshair,

Hides crosshair. You can use "crosshair false" instead, it's easier and doesn't require sv_cheats true.

512=vehicle crosshair,

CS2 doesn't have vehicles that the player can enter. I looked if it changes anything else but it seems to be unused.

Since it is a bitmask, you can hide multiple things at the same time. You just need to add the values of what you want to hide. For example, let's hide both chat and weapon selection (with health, armor, etc) at the same time:

1=weapon selection,
128=chat
Add 1 and 128 = 129

It indeed hides both chat messages and everything what "weapon selection" hides.

Keep in mind that CS2 is still under development so above details may change and there may be something I missed. If it's not going to be too much of a problem please report inaccuracies you've noticed in the comments, and thank you for reading!!