Access to the proofs (evidence by testimonies and documents, etc.) is limited to the parties and their advocates at the time of the publication of the acts. See canon 1598 for details.
Running a risk of over simplification, somewhat . . . This permits them to exercise their right of defense, basically a right to know what the process is and to present proofs supporting their position in the process. But sometimes petitioners and respondents do this out of curiosity even though they “favor” a decree of nullity. It doesn’t matter why. They have the right. Others do not.
This kind of access, inspection at the time of publication, is necessary for a valid sentence. The law does regulate time limits for such actions though. The parties and their procurators also have access to the sentence in order to appeal or raise a complaint of nullity of sentence. See canons 1614 and 1615 in particular.
What is not permitted is third party access. In the US, there is a long history of a First Amendment basis that prohibits the intrusion of secular courts into the internal regulations and tribunals of religious bodies. Even the EEOC has a long list of failures in trying to impose certain standards beyond any that are generally neutral on Church employment when the duties are of a religious nature, but that is side issue.
From time to time, a civil attorney might ponder trying to force disclosure, but none have been successful, to my knowledge. The last federal case, as I vaguely recall (since I have that kind of material back at the ranch), on the particular point was a couple decades ago. The court held that the Church could not be compelled to release material from a tribunal process.
So the issue seems relatively settled, but one never knows these days.