What do British and American post boxes say when they don't want any advertising?

Advertising leaflets shoved en masse into mail boxes are one of the banes of modern society.

In Germany, putting a note saying "Bitte keine Werbung" ("No advertising please") on your box protects you from them - advertisers who repeatedly ignore the note face heavy fines.

Is there a standard English phrase that is widely used in this specific context?

I've never seen such a thing specifically in the US, but the phrase No Solicitation (or No Solicitations) is often posted in public places (or at private businesses) where activities like posting advertisements, handing out fliers, or asking for donations are not allowed.

I haven't seen one for many years, but the standard sign in the South of England used to be No circulars or hawkers.

I think you'd see No circulars or junk mail more often today, but I doubt if people delivering leaflets ever take any notice of such signs.

In Australia it's very common to see a "No Junk Mail" sticker. Less commonly you might see "Authorised Australia Post Mail Only" or "Addressed Mail Only".

The classic version is "no circulars" in the UK, but it's totally ineffective, as there's no legal backing for it.

Advertising flyers - frequently glossed as "pizza leaflets" from the endless supply that pizza firms stuff through every letterbox in their delivery area, seemingly daily - are a common bane of life in the UK, and there's no right to stop them being delivered.

In the US, the issue doesn't arise in the first place, because they don't have letterboxes in the way Europe does. European houses have a "letterbox" (though there is often no actual box) on the front door, and anyone can put something into it. Leaflets from politicians and from local advertisers are common. Newspapers are also usually delivered this way to subscribers. All sorts of delivery services, including the mail, deliver letters and small parcels (e.g. CDs, DVDs, paperback books) this way.

In US, though, you have a "mailbox", which is at the roadside, not on the doorfront. These are the classic ones on a post with a little red flag to indicate that there is new mail. Only the US Mail and the resident can put things in - and the resident can put letters in for the US Mail to collect, unlike the European system of only being able to put outgoing post into a pillarbox. It's a criminal offence to interfere with anyone else's mailbox, for the same reasons that it's a criminal offence to intercept or interfere with the mail in most countries.

I see "no flyers please" and "no junk mail" stickers (and handmade signs) on many people's letter boxes (the personally-owned things you attach to the outside of your house, near the front door) in Canada. I don't know if they work or not. They certainly don't carry the force of the law.