Does Redshift Indicate Mutual Movement Between Distant Galaxies?

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Redshift indicates that a distant galaxy is moving away from our galaxy, which is a consequence of the universe's expansion. Observers in that distant galaxy would also perceive a redshift in the spectrum of the Milky Way, suggesting mutual movement. The discussion emphasizes the importance of frames of reference in understanding redshift and relative motion. Special relativity and the Doppler effect are relevant concepts that help explain these observations. Understanding these principles clarifies the relationship between redshift and the movement of galaxies.
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I don't understand whether this is true or false. It goes like this:
if a distant galaxy has a substantial redshift ( as viewed from our galaxy) then anyone living in that galaxy would see a substantial redshift in a spectrum of the Milky Way galaxy.

I know redshift is when the object is moving away from our galaxy, then wouldn't we see that galaxy moving away as well? am i offtrack??
 
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Yes, we would see the galaxy moving away from us too, but what does that have to do with the question? The question is hinting at frames of reference.

Do you know about special relativity and the doppler effect?
 

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