Learn Turkish: Sadece birbirimizin fikirlerini tartışıyoruz.

The base word is birbiri (each other). Since the sentence is about "we" doing the action, we first add the "our" suffix to get birbirimiz (our each other). Finally, because we are talking about each other's ideas, we add the genitive (possessive) case -in to make it birbirimizin (each other's).

If it is "our" each other's ideas (birbirimizin), why does fikirlerini use the 3rd person possessive (-i) instead of "our" (-imiz)?

This is a tricky but important rule! Even though we say birbirimiz to show that we are doing the action, the word birbiri acts like a 3rd person owner grammatically. So, just like you would say Ali's ideas (Ali'nin fikirleri), you say each other's ideas (birbirimizin fikirleri).

Why is there an extra "-ni" at the end of fikirlerini?

Let's break it down: fikir-ler-i gives us "the ideas of" (belonging to each other). The verb tartışmak (to discuss) requires its specific direct object to take the accusative case. So we add the buffer letter -n- and the accusative suffix -i to point out exactly what we are discussing, creating fikirlerini.

Can tartışıyoruz also mean "we are arguing"?