The parallel isn't exact, but the magistri of the great universities (whether in the faculty of theology or elsewhere) were certainly public figures. And they expressed their intellect in public, as quodlibetal debates were public, albeit in Latin. When a pope or king wanted to assert a position or theology as canonical and orthodox, they often turned to the masters for their opinion, and we have several extant examples of such opinions which were drafted as public instruments.

This latter diplomatic point also informs your question. There is a technical distinction in medieval law (canon and Roman) between a persona privata and a persona publica. A persona publica had a fama, a term hard to translate but is roughly "well-known reputation," and could issue written legal documents on their own authority, whereas a persona privata required the manus publicus (public hand) of a notary or similar to give legitimacy to their action. University masters, particularly those with chairs or other positions within the university, were often personae publicae, and thus were in a very literal sense public intellectuals.

Yes! There were many famous intellectuals in the 12th century who's works and ideas were considered influential and controversial. Perhaps the best example would be Peter Abelard and his intellectual nemesis, Bernard of Clairvaux. The two of them had a rather public feud over Abelards Sic et Non, which attempted to reconcile apparent contradictions within the writings of the church fathers.

Ultimately, Bernard called a council to defend his defamation at the hands of Bernard. Bernard, however, was a no show and instead succeeded in convincing the pope to excommunicate Bernard and condemn his work. You can read Abelard's account of his life and these events in the Historia Calamitatum.

I think that it is also important to note that Philosopher's were theologians at the time. It is also difficult to separate temporal from spiritual matters in the middle ages as so many church functions are tied to secular matters. For example, who gets to nominate bishops? in the 11th century the investiture conflict arose from an assertion that only the pope has the right to invest bishops, even though kings had been doing so for centuries! Some scholars claim that the intellectual battle over the right of investiture kicked off the first "war of propaganda".

If you are interested in reading more I recommend you look at these two works as starting points.

The Investiture Controversy: Issues, Ideas, and Results edited by Karl Morrison

The Investiture Controversy: Church and Monarchy from the Ninth to the Twelfth Century By Uta-Renate Blumenthal