For example, Jisho marks 医者 (doctor) as sensitive, as does the flashcard set I've been using. I don't understand what this annotation means; "doctor" doesn't seem like a particularly sensitive term to me, but maybe that's just a cultural difference.
The formal and official term for a (medical) doctor is always 医師【いし】, and it's the word that should be used in news media and such.
If you ask native Japanese speakers "Is 医者 is less respectful than 医師?", I would expect mixed reactions. Many people would say say it's a pure colloquial variant of 医師. You can safely say "自分の息子を医者にさせたい", "この町にはもっと医者がいて欲しい" and so on.
But I agree that 医者 is more commonly used in negative sentences (eg "医者になんかなるな!", "あの医者に誤診された"), while 医師 sounds clearly nicer in ads (eg "このサイトでは本物の医師が回答します!"). This could be partly explained by the fact that 医師 is formal and 医者 is mundane, but I feel there is more than that. So...it's "sensitive", not derogatory.
お医者さん is a casual, friendly yet respectful way of referring to doctors. People usually don't say "息子をお医者さんにさせたい", but saying "この町にはもっとお医者さんがいて欲しい" is okay.
Addressing to someone directly using 医師/医者 is always rude, and it's no different from calling your teacher 教師 instead of 先生. For addressing doctors, you can always get away with using 先生 (or ～先生).