The first part
It's grammatically correct. Here's what the first part means, with perfect tenses removed:
Recently, I was listening to the music that ArtistX made before he collaborated with GroupX.
That is, have been listening refers to the recent behavior of the speaker. ArtistX before he collaborated with GroupX is a noun phrase referring to music made by ArtistX, possibly many years ago. The present perfect refers to the speaker's listening, not to when the music was made. Probably what's confusing is that ArtistX is being used to mean not ArtistX, but music made by ArtistX.
The second part
The second part means:
My experience includes listening to the music made by GroupX, and I had that experience before I listened to music made by ArtistX.
This is one of those times when English grammar causes conflicts which are hard to resolve in any satisfactory way. (Here's another.)
By choosing the present perfect, the speaker means to point out that his past experience is relevant to his present claim in the first part of the sentence. That is, the speaker's having listened to GroupX before having listened to ArtistX implies something important about the speaker's ability to appreciate or recognize something about the music made by these people at different times. Since the claim is made in the present, the speaker wants the present tense. Since what's being described is something included in an accumulation of experience, the speaker wants to use the perfect aspect.
It goes just a little weird, though, at before I listened to ArtistX. Mixing the present perfect with the simple past here sounds contradictory, although I think it's technically okay. The reason for the apparent contradiction is that "I have listened to" refers to the speaker's music-listening up to the present time, but "before I listened to ArtistX" excludes all time after the speaker started listening to ArtistX. The meaning is clear enough, but the grammar is sending incompatible signals to the listener.
These are also not fully satisfactory:
(1) I've been listening to ArtistX before he collaborated with GroupX, and I had listened to GroupX before I listened to ArtistX.
(2) I've been listening to ArtistX before he collaborated with GroupX, and I had listened to GroupX before I had listened to ArtistX.
(3) I've been listening to ArtistX before he collaborated with GroupX, and I listened to GroupX before I listened to ArtistX.
The problem with (1) and (2) is that the speaker wants to refer to all experience up to now, in order to be relevant to the recent music-listening described in the first part of the sentence. The past perfect agrees with "before I listened to", but not with "I've been listening".
(3) fails to link the speaker's early experience listening to GroupX with the present, but it's probably the best way to resolve all these conflicting pressures. "I listened to" refers to time "before I listened to ArtistX", which is right, and sends no mixed signals. "I listened to" is compatible with "I've been listening to" at the beginning; it just doesn't emphasize what the speaker wants to emphasize.
There are more possibilities, but I think really is no fully satisfactory way to join all these thoughts into one nice sentence in English. Something has to get sacrificed. And the easiest thing to sacrifice is the "accumulation up to the present" in "I've listened to GroupX". The listener will understand it even if the grammar doesn't explicitly point it out.