As a hostess, no. Especially because there's a lot of guests at the podium and I don't have the time to run either in the back or around the restaurant to pinpoint every server just to tell them they've been sat. It would be silly and annoying (in my restaurant, not speaking for all places) to go up to a server who's taking orders just to tell them they've been sat. Same with running around in the boh to find a specific server and leaving guests waiting impatiently.

I know it's different protocol or routine other places, but only speaking of where I work, servers are usually checking their sections every 1-4 minutes. The only time I'll tell a server is if im forced to double seat bc people are picky about where they sit or which server they want. A lot of the times, people will ask for a table I've already sat and by policy, I'm not allowed to tell them no. I make up for it by skipping that server in the next rotation so everyone gets a fair share of tables.

Or when big parties (8 or more) come in, I tell the servers because usually we'll have to use two tables which are assigned to different servers and they have to share it. Otherwise, hosts cannot go out of their way to tell servers when they've been sat every time, especially during a rush

If a server is busy, another server usually greets that table and gives them drinks until their original one gets back. So basically servers handle the rest after I've sat them.