English: He "gives off" a strange vibe. Español: Él se "desprende" extraño.
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It is closer to "give away" In your example, give off, could be translated to "emana" o "genera" English: He gives off a strange vibe. Español: Él emana/genera una extraña influencia/sensación (note: added a suggested translation for vibe, although it is slang, so could be localized and have many options for translation)
You could think that "desprender" is a more conscious act, like "give away" that you chose when to do, while "give off" can happen any time.
"Desprender" is a verb indicating something is getting separated from something else, so it is technically correct, but not close to what the English phrase meant.
Now, speaking of Mexico, "desprender" can be used in the following contexts, along with some of the already listed here, such emanate.
1. Let go
When somebody is truly attached to some other person (perhaps in such an emotional attachment state) - the word desprender comes to mean such as letting go.
"You should let go that relationship man...no good for you".
"Deberías desprenderte de esa relación...no es buena para tí".
2. To remove
This is a most figurative meaning, commonly used in a very specific situation...it tells to remove some part that has been attached into another part, but either accidentally or likely "unwanted"...
"I went through that yard and I got all covered by insects...I gotta get them off me!"
"Pasé por ese baldío y me llené de insectos...necesito desprenderlos de mí!"
Interestingly enough, in Honduras, Central America desprenderse is used to describe the action of a bag of garbage emitting a foul odour.