In the end, they are Latinized versions of either a locally-used name, or another group's name for them. This makes sense, given that the Common Germanic plural ending for Alamann would have been Alamanniz, which wouldn't have been meaningful in a Latin sentence.
Both Common Germanic and Proto-Italic (and thus Latin) inherit their noun declensions from late Proto-Indo-European, but they did so differently. There are clear similarities, but they ended up classifying their noun-stems differently based upon the specific PIE reflex. They both still inherited, partially, the ability of PIE to form different word-types based upon different stem types by affixing suffixes (thus why Latium in Latin provides you with Latius, Latinus, Latialis, etc...) but they also did that differently. I won't go into detail about all the differences as they aren't relevant, but in short, they both inherited the declensions to form plurals differently and also classified nouns differently. There's also a distinction in that, generally, the Latins used adjectives in this case: latini and romani are adjectives (formed by suffixing -[e|i|a|u]nus to the stem noun, forming an adjective marking relation), not plural nouns. In the case of a name such as alemanni, however, it is a plural noun derived from a borrowed noun - alemannus, which to the Germanics was alamanno.
The Romans (and others) also weren't aware of any common ancestry between, say, Latin and Germanic, or any other group (though they believed that Latin and Greek were related; as in Latin being derived from Greek), so they wouldn't try to reconcile it further than either taking the original name and trying to make it fit into Latin grammar, or calquing it.
For the two examples, in particular, that you gave:
Dumnonii: the etymology in the end is disputed though is likely from the root dubn-, meaning 'deep' or 'earth'. Both Latin and Brittonic inherited the PIE suffix -nós, which formed (verbal) adjectives, though it became generalized to creating adjectives. In Latin, this became, as said before, -[vowel]nus, in Celtic this became -inos and thus Brittonic -in, giving us dubnin, pluralized as dubnini. Dubnin is maintained in Devon via a reflex. I am unsure if the Romans had adopted the name of the area and appended their own suffix, or adopted their tribal name and simply maintained it - the result would have been basically the same. If they were to have calqued it, it would have ended up becoming something like Altiuni, though my derivation may be off a bit, and it assumes a specific translation of dubn- as 'deep'.
Alemanni: probably derived from the Common Germanic Alamanniz, which would mean "all men". Common Germanic, as far as I've ever been able to tell while studying it, didn't reduce the meaning of the -inos suffix from PIE, and thus it maintained the meaning of 'made of X' - thus wooden. As far as I've seen, though if anyone knows different I'd love to know, you don't see them suffixing places with it to create adjectives as it wouldn't make any sense. The traditional way in Germanic to form something that refers to a land would be to suffix it with -land, -rikija (which was actually borrowed from Celtic), or in some cases -þeudo ("people", as in the case of Svíþjóð in Old Norse, "Sweden"). You do see cases of a tribe's plural ending up being used to refer to the region they inhabit: Mercia comes from mierce, which is also the plural form of mierce - "the border people" (from earlier OE mearce, the root being mearc, or "march"). The Alemanni were a tribal confederation, mind you, not a single tribe, though this is common amongst many tribal systems (see the Latins, the Saxons, the Alemanni, and so on). They likely would have referred to the land that they occupied, if they had a specific name for it, as alamannolanda, alamannorikija, or just alamanniz, or something derived from their own tribal name. The derivation here could be a bit different depending on how late we're speaking - Common West Germanic conjugation is a bit different from Common Germanic. If Common West Germanic, the plural form would have just been allmanni, and also alternatively allmannoland and allmannoriki. Since they persisted in some form until the Old High German period, at that point you get alman (though later becoming almaner), almannolant, and almannorihhi. I'm having trouble making a reasonably Latin calque of it, it always comes out being awkward at best.
In shorter:
Dumnonii: They would have called themselves, in the relevant time period, something like dubnini and their region something like dubnin.
Alemanni: They would have called themselves, in the relevant time period, depending on a number of factors, something like alamanniz, and their region alamanniz, alamannolanda, or alamannorikija.