From what we know, Honoria and her brother Valentinian III weren't on the best of terms, since he was forcing her into a marriage. One would assume she was being at least under surveillance. Even if that wasn't the case, how was she able to communicate with the empire's greatest enemy without anyone noticing, at least at first?
Hi, I've been meaning to answer your question but I work a lot and haven't had the time. I apologize for the delay.
I like the way you worded this question actually, because this opens up the question of how postage worked in the later Roman empire, beyond the usual "Did Honoria really request Attila to marry her?" question.
The public postal system for the Roman Empire was part of the cursus publicus, which was in many ways similar in concept (but also very different) to the "Pony Express" of the 19th century U.S. This system was likely adopted from the royal mounted courier service of ancient Persia known as the angarium, and was divided into two sections in late antiqutiy: the cursus velox and the cursus clabularius. As a member of the imperial family, Honoria had access to both of these networks. The cursus was by this stage also maintained via a series of hereditary obligations of those living along the routes, so even in areas of the empire that had "broken down" (e.g. Pannonia), it would have still been operating. The patron provided the courier, and those along the cursus were expected to change out horses, repair wagons, provide overnight hospice, and other general services.
To better understand why a private missive like this, from Honoria to Attila the Hun, wouldn't have been intercepted, you have to look at exactly what and when the ordeal with Honoria's "affair" happened. Obviously in ancient Rome women weren't exactly equals, although some upper-class women could on occasion find themselves with vast amounts of influence and power if they were in the right position at the right time. Obviously, members of the Imperial family all had immense power, particularly in religious aspects - Galla Placidia and Honoria were both noted patrons of most of the 5th century basilicae, either directly funding their construction or in the case of Santa Maria Maggiore (the first purely church funded basilica), donating relics. But Imperial women were not supposed to be caught having sex and typically had to be ascribed to a life of chastity - first and foremost due to the fact that they could technically bear heirs and potential usurpers to the Imperial throne, second because of the general expectations of women in Roman culture, third because of the expectations of Imperial women in the new Christian paradigm.
Honoria's affair was not anywhere close in time to Attila's invasion of Gaul, really. It happened in 434 AD when she was still rather young (we don't know exactly how old she was but she was made Augusta in 426, so must have been at least 12 by then, so at minimum she was at least 20). It was discovered because obviously, she got pregnant, and she was ferried off to Constantinople to cover up the pregnancy. Most people did not know this had happened - Priscus who was in with the Eastern Court didn't learn about it until about 449. Only the most influential members in the government like the actual Imperial family, and probably both Flavius Aetius and the Eunuch Chrysaphius, would have known. What Honoria got in trouble for was political scheming and she was divested of the title of Augusta and betrothed to Flavius Bassius Herculanus in 450. This was in all likelihood actually the result of Galla Placidia's death, rather than anything Honoria was actually doing. Valentinian III had somewhat lived in the shadow of both his mother and Flavius Aetius for two decades and with her death he had the ability to assert himself more at court (contrary to popular belief Valentinian III was rather competent and had significant power, see Dr. Meghan McEvoy's work Child Emperor Rule in the Late Roman West for more on that).
However, that isn't to say Honoria wasn't politically incapable; she had spent her life in the shadow of her mother as well, who was a master of court politics. And she could get out of her marriage to an impotent Eunuch by leveraging power in the bureaucracy, but unfortunately no one with that kind of power (Aetius, Ferreolus, Anicius Bassus, etc.) would do it. Attila was an option because he had been granted the title of magister militum (as an honorific, more or less) at some point, probably in 439. So she was really taking a play from Aetius' style, rather than Placidia's, and trying to leverage outside influence that technically actually had legal and military authority in the Roman court. So she sought to use his influence to annul the betrothal.
How she got a missive to Attila basically revolves entirely around the fact she hired a courier, sealed the letter, and nobody broke the seal until it reached Attila. Honoria may have been under "surveillance" but back then the action of "keeping an eye on someone" was far less direct, there were no cameras or such. Influential people at the court would get word of any dealings from lesser bureaucrats and officials, and that took time. Even if she was nearly constantly accompanied by a "male supervisor", or an entourage, there would have been ample circumstance for her to send a missive via a courier. It was sent with a signet ring and a gift in gold, so this probably involved multiple people, and it may have been part of a standard imperial envoy to the Hunnic court. Couriers operated within a Patron-client system, and could be executed for breaching the seals of the missives they carried. Families operating the cursus publicus obviously had similar expectations placed upon them, in order to keep confidential information or important persons secure. Obviously we'll never know exactly how Attila received this missive from her, but in general we can assume it was sent like most other missives along the cursus publicus, where a single courier took the cursus clabularius, or it was a part of larger envoy.
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