The Hebrew Bible is essentially identical in all Jewish communities, despite significant differences in spellings and wordings from the Septuagint or texts in the Dead Sea Scrolls, or even from quoted verses in the Talmud. For example, there are only a few spelling differences between the text of the Torahs used in Ashkenazic, Sfardic, and Yemini communities, despite relative isolation and differences in other customs between them. So this raises the question, at what point were all the manuscripts synchronized across Jewish communities worldwide? At what point were the manuscripts standardized, and at what point or points did these copies make their way across communities?
The story of the standardization of the Hebrew Bible begins in late second Temple times, as evidenced by the Talmud (Ketubot 106a) attesting to scribes that were paid by the Temple to periodically compare the official scrolls to other copies and make corrections. Before this time, it is evident that significant variation in copies existed as evidenced by differences between the Masoretic Text (considered authoritative for modern Jews) and older copies/translations such as the Dead Sea Scrolls manuscripts, the Samaritan Pentateuch and the Septuagint.
Talmudic period: Move towards uniformity
After the destruction of the Temple in 70AD, Rabbinical authority grew in importance as power was no longer centered in the Temple authorities. This growing importance allowed leading Rabbis to exert greater and greater control over the production and standardization of Biblical scrolls. Additionally, many manuscripts were destroyed in the Jewish War, reducing the amount of possible textual variants to pick from.
Rabbi Akiva (c. 40-135) was the leading voice in a movement to create a "perfect text" with standardized spelling, grammar, and pronunciation. Akiva's work was continued by his students, including Rabbi Meir. Over time, methods of emendation were developed; marginal readings were collected and replaced the main text when it made sense to do so. There is considerable discussion in the Talmud about textual oddities from this period.
Masoretic era: final standardization
Beginning around the 7th century, a group of scribes known as the Masoretes grew to be the primary authorities on the text due to their excellent techniques that greatly reduced the chance of scribal errors in the copying process, standardized the paragraph and verse divisions, and added punctuation.
It is also during this period that vowels were first marked in the text, likely due to Syrian influence, vi marks above and below the consonants. Several different systems were created - the "Babylonian" system, the "Palestinian" system, and the "Tiberian" system. The Samaritans also had their own system, closely related to the the Palestinian system. The Tiberian system eventually became the standard and is the only system used by modern Jews. The following chart from A Grammar of Biblical Hebrew compares the different systems.
Minor variations from community to community remained, but slowly the number of versions decreased as communities adopted the text of others. By the tenth century, there were only two versions of any significance left - Aaron ben Moses ben Asher was the main authority behind one and Moshe ben Naphtali was the main authority behind the other.
After their day, very few changes were made. The ben Asher version won out and has been the basis for most manuscripts and printed version since then. However, it wasn't until the 1525 when the first "official" version of the modern text was printed by Jacob ben Ḥayyim ibn Adonijah.