May someone use (through borrowing) keilim that belong to another Jew that have not been toveled yet?
No. Utensils borrowed from a Jew are obligated in tevilah, and may not be used without immersing them in a mikva.
Although borrowed keilim from a non-Jew are not obligated in tevilah, keilim belonging to a Jew have already become obligated in tevilah, and they are not exempted by means of borrowing.
See Tosafos in Avodah Zarah (75b), Rosh (35), Semak (199), and Beis Yosef / Shulchan Aruch (Yoreh De’ah 120).