Are both grammatical？
Yes, Robby zhu, both sentences are grammatical.
Someone says there is a difference in meaning: the first implies you recommend more than one, whereas the second means you recommend only one. Is that true?
Yes, it is true. In the first sentence, "who should be sent to work there" restrictively postmodifies "a man whom you recommend." Therefore, that sentence is saying that, of the men whom you recommend, it is the one who should be sent to work there that the speaker does not quite trust. There is more than one man whom the speaker recommends, but only one of the men whom the speaker recommends should be sent to work there, and it is that man whom the speaker doesn't quite trust.
In the second sentence, both relative clauses are still restrictive, but they separately modify the singular noun. Therefore, only one man is being referred to, namely, the man who is identified by the two properties specified by the two coordinated restrictive relative clauses, "whom you recommend" and "who should be sent to work there." In other words, there is a man of whom it is true that the speaker recommends him and he should be sent to work there, and that is the man whom the speaker does not quite trust.