It depends on how the spell actually works, on which there is not a consensus
Wall of Fire is poorly written and can be interpreted to mean that creatures on both sides of it take damage. However, everyone1 agrees that there is a side on which creatures are damaged, and a side on which they are not. At best, one can say that the spell can be interpreted to yield the results of how it is actually ruled. At worst what it actually says, RAW, contradicts the way it is played. While unfortunate, this contradiction is not usually a problem - since the spell can generally be used in play without us having to agree on how and why it works that way.
However, in the case of the OP's problem, we actually do need to know how the spell works. The OP is positing a situation of a Wall of Fire cast as a ring of smaller than maximum size, damage side inward, with 'friendly' creatures close at hand outside. In this case, the friendly creatures are on the non-damaging side of the wall, but they are also within ten feet of the damaging side and in the 'direction' of the damage. To rule on what happens in this case, we do need to know how the Wall of Fire works and why it works that way.
In the aforementioned question about how to rule Wall of Fire there are currently three net-positive answers, and they all agree that the Wall has a side on which creatures take damage and a side on which they do not. However, they disagree in their explanations for that ruling.
Vylix makes an argument of intent. Since a ringed wall with damaging-side-out is meant to protect those inside it, it cannot be the case that those inside would be damaged even if they were within ten feet of the damaging side, regardless of what the spell actually says. To extend this logic to the current case, a ring with damage pointed inward is intended to damage those inside the ring and not those outside, and the spell explicitly permits one to cast it at a smaller than maximum size. If the damage from the damaging side could pass through the far side of a smaller ring, it would damage those outside it, which clearly goes against the intent of the spell. Thus, the damage does not pass through, because it is non-sensical for the spell to act in a way that contradicts its intended use.
Rubiksmoose's (accepted) answer makes an argument of common use.
They admit that it is possible for a RAW reading of the spell to result in damage on both sides, but quickly discard that as not being the way the spell is played in practice. They conclude that "RAW should be disregarded here in favor of the more common and natural reading of the spell, but your table should do whatever is the most fun for them." Unfortunately, that doesn't help us resolve the OP's question. Since MivaScott is asking about the specific and uncommon case of a inward tight ring, there is no "common and natural" resolution. One could presume that "what is the most fun" is for the spell to work the way the caster wants it to, though.
Seidr makes an argument of mechanism. They assume that the RAW description of the damaging side of the wall affecting creatures within 10 feet is correct as written, but reason that even if the damaging side of the wall radiates damage in all directions, the spell is, after all, a wall, and thus blocks the damage from coming through to the non-damaging side. In this interpretation, creatures outside a tight ring would not be damaged, because the same ring would block the damage from passing through.
Kirt makes an argument of definition. He explains that the wording of the Wall of Fire spell reveals that it fits the description of having its area of effect in the shape of a line. In this interpretation, the damaging effects radiate unidirectionally from the damaging side of the wall. Following the rules for Areas of Effect,
A spell's effect expands in straight lines from the point of origin. If no unblocked straight line extends from the point of origin to a location within the area of effect, that location isn't included in the spell's area. To block one of these imaginary lines, an obstruction must provide total cover.
Checking the rules on Cover, we find that
A target has total cover if it is completely concealed by an obstacle.
The 'completely concealed' part is easy; the wall is opaque. Unfortunately, the PHB does not define what is an obstacle, and the examples it provides are only given within the sections on half and three-quarters cover; there is no example given of what might provide total cover. Thus it falls to the DM to decide whether the wall of fire serves as an obstacle which will provide total cover from its own damaging effects should it block the area of effect, and thus whether it would protect those outside a tight ring.
Setting aside the referenced question, in an alternate answer to this question itself, WakiNadiVellir also makes an argument of mechanism, but with an opposite result to that of Seidr. They say explicitly that the wall does not block damage from the damaging side, and implicitly that the damage radiates from the damaging side in one direction only (in an earlier comment they compared this to an image radiating from the surface of a mirror). Under this interpretation, a tight ring would damage those close outside it, since the damage radiating from the damaging side passes through to the non-damaging side.
Everyone1 agrees that there is a side of the wall of fire on which creatures are damaged, and a side on which they are not - but how that is supposed to work is clearly in dispute.
The OP's question requires an understanding of how the wall works in order to be answered. It is impossible to answer the OP's question without determining specifically how the wall works, which is where RAW fails for the wall. Thus, there is no RAW answer to OP's question, although they are free to choose some other criteria for an answer.
1Well, not everyone rules that way. In the referenced question about how to rule Wall of Fire and whether or not one side is safe there is an answer arguing that the spell as written is correct and that creatures on both sides of the wall are damaged. Currently that answer has only downvotes and is deleted by the author. Perhaps most importantly, MivaScott, the OP of this current question says "that is not how any DM I know rules it", so we can assume a one-sided ruling when answering this question at least.