(You might want to read the Tzvi Lampel’s translation of the Rambam’s introduction to Pirush HaMishnayos, and his books the Dynamics of Dispute). But in short the answer is as follows:
When we say that Torah Shel Bal Peh was given from Hashem to Moshe we mean different things depending on the subject. So there are some halachos that are Moshe m’pi HaGevurah as it is understood simply (e.g. the color and shape of teffilin). There are other parts of the Oral Tradition that are received understandings/interpretations as to the meaning of words in the Written Torah (e.g. that “an eye for an eye” refers to a monetary payment). In these areas there can be no dispute in principle (one cannot argue on what Hashem told Moshe) but only a dispute as to what the tradition actually is. Therefore dispute in these areas arises only because somebody forgot or made a mistake in the chain of tradition.
Moshe Rabbeinu was also given a series of methodologies (like keys to a code) that would allow the Chachamim to derive new halachos when faced with new situations. It could also be used to revive forgotten halachos. On these, in each generation a given Sanhedrin could argue and reverse the decisions of a previous Sanhedrin and therefore the Halacha could change. Of course human beings can have machlokes in the applications of these methodologies. While in earlier generations these machlokesim could be resolved by the Sanhedrin, in later generations when there was no regularly sitting Sanhedrin, machlokes could proliferate.
Of course there were also many disputes about whether or not to make gezeiros and takanos, and then later about the extent of those gezeiros, their applicability to new situations, etc.