While the saying sounds quite interesting and intriguing, I am not aware of any source for it. In fact, it doesn’t even sound right according to Jewish teachings. Judaism teaches that every person is always responsible for every other person, to save them financially and to be there for them with acts of kindness and any assistance necessary.
If anything, the attitudes of the saved and the savee should be as follows. The saver should be thankful to God for the mitzva of saving a person’s life. The savee should be thankful and grateful to the saver for saving his life, and express his appreciation to that person and to God.
There is a Mishna found in the Babylonian Talmud, Tractate Sanhedrin 37a that reads, "Anyone who destroys a life is as if he has destroyed an entire world and anyone who saves a life is as if he has saved an entire world..." Perhaps the quote you cite has been adapted from that as follows. Whoever saves a life is responsible for “it” – meaning it and the entire world being saved and continuing to exist.