I've often seen the tribes of barbarians the Romans fought be anywhere from a few hundred people in a single town of wooden and straw huts, to about a dozen towns, each containing over a few thousand people, with more permanent city structures. Considering that the population density would be lower than a Roman province, how many people, including women and children, would be part of a tribe? For reference, I'm asking about larger and more well known tribes such as the Arverni, Angles, Cherusci, the Burgundians, etc.
I'm going to take this question in two parts. First starting with the idea of "tribes" and then moving on more directly to answer your question. Also I can only really speak about the groups that were present during the collapse of Roman authority in the western parts of the Empire and into the early Middle Ages so please bear that in mind.
The idea that the various groups that lived outside or on the periphery of the Roman world constituted "tribes" is a long standing convention and is reinforced by the contemporary writers of the time. However it is deeply flawed for a variety of reasons. Foremost among the reasons is that the word "tribe" carries certain connotations to the modern audience that you touched on in your question, implying that the people of these areas lived in wooden and straw huts and were far more primitive than their contemporaries in the Roman world. The other problem with your question is the assertion that the Romans and the Celtic/Germanic populations only fought each other. These assumptions are simply not true.
These peoples maintained advanced trade relationships with the Roman world, had their own complicated internal politics, and produced their own art and material culture (of course with other goods coming in through trade). They did not have the same level of intensive agriculture or trade or manufacturing (for want of a better word) as the Roman world, but it was still a sophisticated culture.
Again, I can only really speak to the Germanic peoples, but trying to force the various groups into tribal names, some of which were in use for centuries and almost certainly do not reflect the reality on the ground. The labels have stuck around because they are extremely useful shorthand. After all it is a lot easier to talk about the Vandals, Ostrogoths, and Visigoths as monolithic groups as opposed to groups that were constantly shifting in their allegiances, relationships with each other, the Romans, and so on. This is also borne out by the ancient sources that we have. They were more than happy to ascribe labels to these peoples as singular entities that coalesced to threaten the Roman world, but it is a gross oversimplification.
Furthermore, many of these people would have considered themselves Romans, many of them fought for the Roman Empire. Tying them into one distinct identity is an oversimplification. There was nothing inherently incongruous about being from a Germanic group and a Roman.
Now that we've gotten that out of the way, let's look at the numbers part of your question.
How many people constituted a "tribe" is almost impossible to nail down with exact certainty, but it is possible to try and reach some broader conclusions.
First off, we can know that the level of agriculture was not as advanced as the Roman world so a lower population density would make sense. However given the large areas that these people inhabited, total populations would have still been quite large and important sites from the time period have had estimated populations into the thousands. However that does not give us a picture of the population of each group, just what the larger geographic areas were capable of sustaining.
Contemporary (Latin and Greek) sources, often seem to talk about various tribes massing together and fielding armies that numbered into the 10,000's of men. While I am less familiar with the scholarship around it, I know that Caesar's accounts of the Gallic Wars also include extremely large numbers of enemy warriors coming together.
Now the question becomes how do we explain these numbers? Are these inflated numbers by biased sources? Are they reporting total numbers of men involved including camp followers, and in the "migration" era, families and other hangers on of the warriors and leaders?
It is perhaps possible that these numbers refer to able bodied men bearing arms but it seems very unlikely. Warfare among these groups was largely the purview of the aristocratic classes, the local kings, chiefs, and their various retinues of warriors and these were tiny portions of the overall population. The majority of people in a population would not have been mobilized for warfare. This is due to certain practices, women would not have been a part of the armed bands, and many people who would would have been part of these societies would have been slaves and unlikely to have been allowed to bear weapons. Other parts of the population are also unlikely to have constituted warriors, such as the elderly, the young, and the extremely poor.
It is also extremely unlikely that when these people started moving across the continent, and across continents in some cases, that they uprooted their entire societies and went on the road with all their children, women, and so on in tow. Now I will admit, this is not a universally held idea and the debate continues on this subject. While the idea of mass migration at the end of the Roman Empire has been largely called into question, certainly the aspects that relate to the supposed wholesale slaughter of the native inhabitants and the total collapse of Roman society, I do not think the debate has been firmly settled as to in what numbers the Germanic peoples crossed into Roman territory with.
However, it is more likely that the numbers of people that constituted these "tribes" would have numbered maybe into the low thousands at the upper scale. That would encompass quite a large grouping of warriors to be fielded by the Germanic kings and chiefs at the time. Many more of them would have likely been only a few hundred strong. This would be consistent with the political structure of these societies at the time, they were far more loosely organized that we often give them credit for. This is in part borne out by the Roman sources. Tribes often appear in one account only to have disappeared a century later.
We run into more problems when we look at some primary sources that talk about these tribes and their numbers. The story of the coming of the Anglo-Saxons to England is a good example. According to the story as relayed by Bede, the Angles showed up merely in three ships but then later grew massively from continued ships of people coming from their homeland to Britain. He does not give exact numbers, but he proposes that the numbers were massive and they destroyed the native population through conflict or by forcing them to flee across the island. Now Bede's account is extremely suspect and cannot be taken at face value, and there have been studies done (also not without controversy) on the genetic make up of modern Britons that have given mixed answers.
However this account may have a kernel of the truth in it. The initial force of the invaders might very well have been rather small. Bede says that the Angles were hired as mercenaries for the British (consistent with the idea that these groups were typically somewhat small and well armed groups) and that only later did large numbers of people cross over. Now it may be somewhat problematic to pick and choose what we should accept as possible from a source, but these parts of his account seem to be in line with the idea that these moving bands of people at the end of the Empire were relatively small groups that did not organize entirely into large scale tribes that behaved in uniform ways.
In short, its impossible to know the exact numbers of the people who composed the "tribes" that the Romans interacted with. However at the tail end of the Empire it seems that they referred to relatively small bands of warriors and not to massive groups of entire populations that have uprooted entirely.
Sources:
I have drawn quite extensively from the first few chapters of Peter Heather's Empire and Barbarians and while his work is not universally accepted and not without debate, I believe that the parts that I was drawing from are reasonable in their conclusions.
Peter Brown World of Late Antiquity AD 150-750
Adrian Murdoch, “Germania Romana"
Bede, Historia Ecclesiastica Gentis Anglorum
Further Reading
Tacitus, De Origine et Situ Germanicorum, this is where a lot of misconceptions come from, down to today and as well to the ancient authors, but I do think it is worth reading.
Early Germanic Literature and Culture, ed. Brian Murdoch and Malcolm Read, the article from Adrian Murdoch is in here and there are other articles in there that might be of interest.