As I wrote in the comments, there were definitely other centers of Torah around the world.
We find that in Rome there were a number of sages:
Todus Ish Romi (Pesachim 53; there's a disagreement about when he lived. Some push him as far back as the time of Rabbi Shimon ben Shetach).
Rabbi Matyah ben Charash who opened a beit midrash there (Sanhedrin 32b) and had discussions with sages who visited Rome (Yoma 86a, Yoma 53b, Meilah 17a).
Rabbi Yehudah ben Bateira (the second or the third) was born there (Yerushalmi Sanhedrin 41a).
Paltion Ish Romi (Shir Hashirim Rabbah 8:5).
Also, see on the bottom of this essay a list of inscriptions of archsynagogoi (ראש בית כנסת, head of the synagogue) from places around the Roman Empire.
As a friend of mine put it, unfortunately, the most likely explanation for why we don't have texts from these communities is that they simply didn't see a reason to record their studies, for the exact same reason that not every rabbi has ever written everything he has ever thought up of (of course, it is also entirely possible that they did actually write and what they lost was never mentioned in our sources and was eventually lost, like how there may have been other parts of the Yerushalmi). Note that the creation of the Babylonian Talmud was actually an out of the ordinary event, as most Tannaic/Amoraic texts that we have are from Eretz Yisrael. Ultimately, it seems that all Torah centers in the world looked to the Yisraeli center for guidance.
The first hints at a split between the diaspora and Eretz Yisrael may be seen when we find mentions of halachot of the Babylonians and mishnayot of Babylonians (see my recent answer here). This chasm widened as the Babylonians sages continued to become more independent over the course of the Amoraic period. Finally, we find the harsh disagreements between the Babylonian Geonim and the Yisraeli Geonim - so harsh were some of these disagreements, that at times the Geonim of each group actively undermined and minimized the rulings of the other group (I don't have time right now to gather the sources, but see here for a summary).
I say this to point out that it seems that it was only at the end of the Tannaic period that we see some hints of a split, and even then it was only between Babylon and Eretz Yisrael. As far as we know, the other communities continued to view Eretz Yisrael as the more authoritative center of Torah leadership. In the past I've seen studies that point out that in the early Rishonic period, most European communities were more closely aligned with the halachot of Eretz Yisrael, so we see that in Europe Eretz Yisrael held dominance for many centuries, while the Asian and African communities eventually became aligned with the halachot of Babylon as that center became dominant in the Persian and Muslim empires. I'll bring sources when I'll be able to.