Muhammad () is considered a prophet and messenger by Muslims and therefore we (Muslims) honor him by saying:
صلى الله عليه وسلم
Allah's prayers and blessings be upon him (the correct wording may differ a little)
Peace be upon him
some use the abbreviation PBUH -after writing his name, however many scholars say that such things are rather disrespectful.
Any of both options usually is quoted after quoting his name.
However you may find some deviations from that, with the same meaning like:
عليه الصلاة والسلام
(Allah's) Prayers and peace (or blessings) be upon him.
This is done because Allah our Creator ordered the believers to do it:
Indeed, Allah confers blessing upon the Prophet, and His angels [ask Him to do so]. O you who have believed, ask [ Allah to confer] blessing upon him and ask [ Allah to grant him] peace. (33:56)
I don't know of any major deviation or difference between sunni and shi'a about this.
However to be clear I'm answering from sunni perspective.
Disbelievers or non-Muslims are not necessarily addressed by above verse that's why if a non-Muslim would call him simply Muhammad or the prophet Muhammad -which would be more respectful- etc. it would hardly make a difference. Nevertheless a respectful variation would certainly be more appreciated.