Neither.
sattānaṃ belongs with tesaṃ-tesaṃ. And -ānaṃ ending already declines satta and so it is a possessive case of satta. If you're asking what do the sattā possess, then you might say jāti, but that is not exactly right. This is because sattānaṃ doesn't have a standing in this sentence without tesaṃ-tesaṃ. When such declined pronouns occur together, it means "respectively".
By the way, the order of words has nothing to do with the meaning in Pāli and Sanskrit. You have to understand which words belong together using kārakaṃ. For example all of these are exactly equivalent:
sabbe sattā laddhasampattito mā vigacchantu
sabbe sattā mā laddhasampattito vigacchantu
mā sabbe sattā laddhasampattito vigacchantu
sabbē sattā laddhasampattito vigacchantu mā
sabbe laddhasampattito mā sattā vigacchantu
Since we have 5 unique words here you can get a total 120 permutations, and all of those sentences are exactly equivalent in meaning!
So the better thing to do is to look for the case and group words of the same case together. This is a strategy that works if you don't have "compound/complex sentences" ; where two separate sentences are joined together into one. You have such cases if someone is quoting someone else's statement, or if someone is describing two separate incidents in one sentence. Aside from those scenarios, (especially in this case) you can go by simply grouping words by their case.
So:
yā goes with jāti that birth
taṃhi-taṃhi goes with sattanikāye in those respective realms
tesaṃ-tesaṃ goes with sattānaṃ of those [respective] beings
The pattern is the same: the left column has the pronouns, the right column has the nouns, and their cases match.
One last note: when pronouns are repeated like that, they are actually not separate words, they are actually treated as a single group. The repetition has the meaning of "respectively". This is an implied meaning.
So hope all that helped and wasn't more confusing.
Thanks,