Sorry for the late reply! I decided to take a break from actual legal work and answer this very interesting question.

In short: Justices were always permitted to issue dissenting opinions, but Marshall had a remarkable ability to create consensus (including the admirable ability to let his own preferences subordinate when necessary), so this was relatively rare during his tenure, at least compared to today. Disagreements were largely handled behind the scenes, in conference or through correspondence. If a Justice did dissent, there were no real consequences. Lifetime tenure for Supreme Court justices makes them highly insulated from the consequences of their decisions, and as strong a figure as John Marshall was, he had only the power of persuasion over his colleagues. Furthermore, dissent is a dissent: it has no legally binding value, so the consequences of issuing one are mostly academic. And finally, the Supreme Court was (and is) a minuscule, rarefied institution, defined by its exclusivity. Marshall presided over at most 6 other Justices during his tenure, and by all accounts they were close to one another. They came from similar backgrounds and education; were part of a nascent institution fighting to assert itself in the separation of powers; and spent a great deal of time alone together deciding the most important legal issues of the day. They, by and large, got along well, even if they disagreed.

John Marshall instituted the tradition of a "majority opinion," in which the Court spoke with more or less a single voice on a case. Before, the court would issue so-called "serial opinions," in which each Justice issue their own opinion, and lawyers had to read all the opinions to figure out what the controlling law was (based on which legal propositions garnered majority support on the Court). And in the very early years of the Court there was no official Court reporter who issued an official version of the Court's opinions; judicial opinions were oral, not written! Supreme Court reports were not officially funded and sanctioned by the federal government until 1817; many early court reports were criticized for inaccuracy and lateness. (This creates a whole separate issue of relying on early Supreme Court cases as precedent, authority, and evidence for early American legal thought; but that's a topic for a different time).

The serial opinion method was inherited from the English tradition. The advantages of the single majority opinion are obvious: the Court spoke with more authority when it spoke with one voice, especially in an age where the Court's authority in the division of powers was ambiguous. (The Constitution famously does not explicitly grant the Supreme Court the power of judicial review; John Marshall asserted it in Marbury v. Madison, and had there been dissents from that opinion, who knows if the Court would be what it is today?)

Marshall, for various reasons, including leading ideologically sympathetic judges, sheer force of personality, and the so-called "judicial crisis" (in which the largely Federalist court came into conflict with Thomas Jefferson's Democrat-Republicans), was able to create remarkable consensus on the Court, which allowed him to institute the majority opinion. Dissents were relatively uncommon during his tenure. However, they were not unheard of. William Johnson, appointed in 1804, issued a great number of dissents during his tenure (which lasted 30 years). Marshall himself issued a dissent in the major case of Ogden v. Saunders, 25 U.S. 213 (1827).

As far as the historical record shows, disagreement on the Court was largely hashed out in conference (i.e., discussion among Justices); if it survived into written opinions, it does not appear that anyone took great offense. Most of the dissents opened with apologies by the authoring Justice to their fellow colleagues. Marshall himself seemed quite magnanimous in instances where he was dissenting. In Ogden, he opens by saying

"It is well known that the Court has been divided in opinion on this case. Three judges, MR. JUSTICE DUVALL, MR. JUSTICE STORY, and myself, do not concur in the judgment which has been pronounced. We have taken a different view of the very interesting question which has been discussed with so much talent as well as labor at the bar, and I am directed to state the course of reasoning on which we have formed the opinion that the discharge pleaded by the defendant is no bar to the action."

Id. at 332. Hardly reminiscent of the self-righteous indignation of a Scalia dissent, or the cutting moral anger of Kagan. When writing to Justice Joseph Story about his disagreement with him on a case, Marshall wrote: “Should the court concur with you, I shall be far from regretting it for my opinion in this case is not one of those in which I feel such confidence as to regret its not prevailing.” Again, not exactly the fire we have come to expect from dissenting Justices. (Granted, Story and Marshall got along particularly well).

A disagreement with John Marshall might well end up the majority opinion, too. Marshall was quite willing to compromise, listen, and go along with the views of his colleagues. He was deferential to others when he believed them to have expertise. He was willing to "subordinate his views for the sake of uniformity," and no ideologue who insisted on speaking his mind even if it made no difference to the ultimate ruling.

To sum it up: John Marshall's Court was a small, rarefied institution of men who largely respected each other. Disagreements were largely resolved behind the scenes, and on the rare occasion they spilled into dissenting opinions, it did not generate much tension between the Justices. Marshall did not have the power to punish his colleagues to force unanimity; instead he generated unanimity through consensus building, negotiation, and deference.

Sources:

Charles F. Hobson, Defining the Office: John Marshall as Chief Justice, 154 U. Penn. L. Rev. 1421 (2006)

Ogden v. Saunders, 25 U.S. 213 (1827)

Denis P. Duffey, Jr., Genre and Authority: The Rise of Case Reporting in the Early United States, 74 Chicago-Kent L. Rev. 263 (1998)

Erwin Surrency, The Beginnings of American Legal Literature, 31 Am. J. L. Hist. 207 (1987)

Saikrishna Prakash, Imperial from the Beginning: The Constitution of the Original Executive (2015)