Obviously there have been a lot of different types of sieges of different kinds of things in human history. There's no real point in defining the full depth and breadth of that space, since it covers such a gigantic variety of structures, naturally fortified geographical features (hills, peninsulas, etc), and styles of warfare.

Generally, though, some features of subsets of that space that could provide an avenue of communication for aid from outside forces are as follows:

First, it's important to point out that sieges were not a surprise to either side. It takes time for the attacking force(s) to approach and surround a fortification and that time allows the defender(s) to proactively send information about the attacking force to nearby allies, often asking them to act as a rely to farther allies or to higher authority. This could take the form of a runner, a rider, or in the case of coastal cities, quick boat traffic. I'm sure you've heard about some of the common stories in Western history about these messengers (not all of whom were warning of a siege necessarily, but that's a matter of the substance of the message not the tool to spread the message). People like Paul Revere in the US Revolutionary War (not to mention his compatriots who were also charged with a similar message) are a prime example of this.

Second, not all places that can be sieged are vulnerable to complete circumvallation and interdiction. Port cities are notorious for this, in that they cannot be completely cut off and can hold out for ludicrously long times. The Siege of Drepana is an example of this from the First Punic War^^1. After a naval defeat at the outset of the siege, Rome took eight years to rebuild their Navy to return to finish the job. This kind of incomplete siege happens a lot in history, particularly when it comes to stopping individuals or small groups from entering and leaving a fortification.

Third, often the only thing a besieger needs to do is stop bulk supplies and large groups, the small stuff doesn't matter. So a knowledgeable local can make his or her way into and out of the surrounded garrison with ease. In particular, if the sieger is concerned about being attacked from behind, it becomes dangerous to spread your attacking forces all the way around the defender. Julius Caesar's Seige of Alesia is a prime example of this: after the defending Gauls were able to send cavalry out through the incomplete siege works, Caesar was forced to build a second set of outward facing fortifications to protect his inner ones. He was then, in turn, besieged by the allies of the original defenders. He ended up winning the ensuing series of battles and ending that particular conflict in his favor; but the point that sieges are potentially very dangerous because of those escaping messengers is plain.

1. Thiel, Johannes Hendrik. A History of Roman sea-power before the second Punic war.(1. publ.). North-Holland Publishing Company, 1954.

Sieges throughout the whole of history have always been generally a rather good tactic for countering an inferior enemy’s fortified/strategic position, successful sieges excel at stoping most if not all supplies flowing in and out of whatever or wherever is being sieged.

However much like an encirclement these tactics are not flawless, Ill prepared sieges may have exploitable floors, the attackers lines may have gaps, allowing messengers such as riders or even runners (or really quick rowers) to brush past the enemy and go get reinforcements.

If the attacker is well prepared, and effective at standing in a circle around your city, depending on the ability and size of the defending force, an effort could be made to temporarily break the siege, allowing supplies/reinforcements and even communication to flow for a period of time.

However this is more of a modern concept due to how much co-ordination would be required to off such an ambitious manoeuvre, so much so that you probably would need a radio at the very least, and by then you don’t have a communication issue... Another thing to remember is if you’re only attempting to open up a gap for a messenger that may or may not yield reinforcement, it may prove foolish to definitely sacrifice several good men for something that could be achieved via infiltration.

Islands are naturally easier to defend, you can’t march across the ocean (or deep lakes), and attacking forces getting out of long boat could be made short work of by the smallest of armies due to how much of a disadvantage that would place them in. So naively blockading an island would be a better option, or possibly using both navel and ground units to cover all possible escape routes depending on the position of said island.

But the principle doesn’t change, if you can’t sneak out communication, you may need to force it out, and if you can’t force it out you may need to look to other means.

A few less popular choices: -If you were lucky enough to have messenger birds, you wouldn’t have much to worry about other than the bird getting lost, this is probably the safest mode of communication, however is not as widespread as modern media would have you believe.

-You could attempt to draw attention to the siege in an attempt to attract friendly forces, although extreme, setting several surrounding fields ablaze wouldn’t hurt, you can’t farm them anyway thanks to the enemy...

-Wait... yes wait, if you are fortunate enough to be in friendly territory, you can hope someone notices the blockade before being captured/killed by it. And reports the suspicious activity to the local garrison.

Although there are more obscure ways of contacting potential allies, these are the ones most likely to work, actually writing down ’we need help’, or sending someone to communicate the words ’we need help’ isn’t guaranteed to work, and even this is choppy due to the fact it relies on not being intercepted by the enemy.

Related question: How long would it take a typical sieged city to run out of food?