I don't think "according to me" is a natural thing to say in English. You'd only say it if you were translating poorly from another language. "In my view", "In my opinion", "For me" and "I think" are all better for "Laŭ mi".
'Laŭ mi' is found in what languages beside Esperanto?
I believe you can say "Laut mir" in German, though I cannot speak to how common it may or may not be in speech.
It is quite common, there's nothing wrong about using it, you can even use it with other personal pronouns or persons: "laut dir", "laut ihnen", "laut dem Zeitungsartikel".
You can certainly say 'Secondo me' in Italian, which is directly equivalent. I hear it a lot.
Laŭ seems more or less identical to "Según" from Spanish
'In my opinion' was accepted, too
"Vetero" comes from the German "wetter".
"Per me, the weather is good" is just unnatural English. "According to me" I can understand, per me is just ungrammatical.