The most important thing to consider when speaking of sending one to Siberia and the prospect of escape was the capacity one has been sent there. I wrote a brief summary of the exile into Siberia in this thread, so I'm linking it here to not repeat substantial part of it here. The actual ability to escape the punishment was chiefly dependent on whether one has been sentenced to a forced settlement or hard labour (rus. katorga).

In the former case, escape was no easier than in most prisons of or closed penal colonies of the era. Prisoners sentenced to labour were usually sequestered in the facilities guarded by armed personnel, quite often special detachments of the military and were often shackled, sometimes permanently, in a way that allowed work and slow movement but prevented running, making escape very hard if not impossible. In addition, many prisoners sentenced for more serious crimes, especially in 18th and early 19th century could have been additionally mutilated, having part of ears of a tip of the nose cut off, clearly marking them as convicts and making prospective escape even harder.

On the other hand, being sentenced to a simple 'forced settlement' gave a lot of opportunities to escape. As you correctly surmised, the main idea of such form of the exile was to move potential troublemakers from the areas where they could have conspired with like-minded individuals, usually to villages of small frontier towns located thousands of miles away from the large cities and cultural centers. This also meant that authorities did not really cared what the exiles is doing there and if they decided to move abroad, to China, India or Mongolia, they were indifferent to the fact, as the main goal has been already met - the sentenced person was away from Moscow, Petersburg or any other city.

Some exiles we additionally forbidden from moving to the large cities (sometimes only to Moscow and Petersburg), but were otherwise permitted to travel freely. Incidentally, being exiled to Siberia usually meant that people were not additionally monitored by local police, what was usually the procedure used in case of simple exile from the major cities, what happened to Alexander Pushkin, who, exiled from Petersburg and Moscow and banned from moving there, was living for some time in Kishyniev and travelled around Crimea and Caucausus on a provision that he had to present himself to the local police officials, announcing his arrivals and departures.

A good insight into the escape from Siberia in late 19th century can be found in the memoirs of Lev Trotsky who, sentenced to lifetime resettlement to Obdorsk (now Salehard, Russia) managed to escape in Berezovo. He managed to do it by befriending a local man, Kuzma Korovin, son of a Polish insurgent exiled to Siberia in 1864 after the fall of January Uprising, who helped him in escape. Kuzma procured a sleigh with a driver and even a small diversion to sent the potential pursuers in the opposite direction. Trotsky noticed that he could have escaped either along the road or through wilderness, and have chosen the latter, fearing that the road will increase chances of meeting soldiers and the local telegraph line can be used to notice other posts about the presence of the fugitive. Cross-country route, on the other hand, would make the travel more difficult and somewhat slower, but chances of being captured were minimal. Eventually it turned out that no one was chasing him and according to Trotsky's own words, the worst problem was apparently the driver's alcoholism.

One of the most important problems for the prospective fugitives was the sheer distance they needed to cover to get basically anywhere. This was especially important problem before mid-19th century as the relative lack of any railway infrastructure and only rudimentary road network made the travel very difficult and given the harsh climate of the northern part of Siberia and very cold, snowy winters anywhere in the region an attempt of escape was borderline suicidal.

Rufin Piotrowski, Polish officer sentenced to hard labour in Siberia for conspiracy against Russian government in 1846 noted in his memoirs that depending on the place of exile, one could theoretically escape eastwards, to Okhotsk see and then find a sea route to United States, southern and southwest route through Bukhara and Persia and then to India or Ottoman Empire respectively and the western route, through Siberia and Ural Mountains to Europe. Piotrowski eventually decided to use the latter, although he preferred to use side roads linking local villages and smaller towns. By pretending to be someone from the next largel place of destination, he was hiring transport, effectively covering the distance from Omsk to Irbit through Tara and Tyumen (850 km) in the first three days. After having part of his money stolen, he later had to cover large part of the way on foot and occasionally on sleigh, moving through Alapayev to Solikamsk and then, through Cherdyn, Kay (now Koygorodok) and Lalsk to Ustyug, what has taken almost two months. In Ustyug he found a barge that carried him last 500 km down the Northern Dvina to Archangelsk.

His account give an interesting, detailed insight into what the escape from Siberia would have entailed in late 1840s. It is worth noting that he painstakingly prepared for his journey, taking measures that would easily befit any modern intelligence agent. During almost eighteen months of imprisonment, Piotrowski observed customs and clothing of the locals, slowly collecting a typical attire and assuming the looks of the Russian peasant (in natural or artificial way, as although he grew an impressive beard, he decided to wear a wig). He also saved some money, managed to procure low-quality fake documents (noticing that finding someone able to make one was easier in his place of imprisonment than in his area or origin, thanks to the presence of well-educated and skilled artists among political prisoners). He also studied the information concerning geography of Siberia and locations of new settlements to plan his journey accordingly and be able to give believable answers when asked by locals or authorities. This required a significant dose of education and general knowledge, something a that was far from a given for a common member of a lower classes in Russian Empire.

In addition, one of the problems that fugitives were facing was the need for the aforementioned documents. They were not really that necessary in towns and villages, where exiles could have been passing for locals, but if they were caught somewhere in the wilderness without the proper paperwork (and the documents issued to people wanting to move from their place of residence hail back to the Act of 1721 issued by Peter the Great) they could have been arrested and brought to their alleged home town or village for further investigation, usually under suspicion of being prisoners, criminals, deserters or fugitive peasants. People seeking help in the villages could have also been asked for documents by locals who, even when illiterate, knew that fugitive prisoners usually don't have any. Please note that Siberian villagers were usually not aware of the political event is the European parts of Russian and most often than not simply assumed that any fugitive is a regular criminal who might be dangerous and thus not worthy of help.