Early on I tend to have dwarves multi tasking, I mean I have to dig out a fortress, gather wood, build tables/chairs/beds/barrels/bins/etc/etc, I have to farm and produce food (that one I tend to let linger a bit as I can usually make it to the first caravan without issue and can make up any difference with plant gathering/fishing/butchering at the last minute), building some kind of defense (I use the fort defense mod which can lead to early sieges, so getting some defenses (walls/soldiers/etc) up early is a must), haul all my stuff inside to safety, producing sellable goods (no more selling large serrated discs for a fortune, not only does their price seem fixed but metal is much rarer now), and if I'm in a climate that freezes getting some underground water source (in case I run out of booze and/or somebody gets injured).
And that doesn't include any optional things I may want to make, or embark specific issues.. So as I said, multi tasking, only high priority jobs get focused on (early on would be mining for example, in a dangerous biome might be defenses as well). I do love how you can use the manager to limit who can work at a workshop, so for example I can only have a few people actually use the mason workshop while others work on outer walls.
That said, once things get going and I don't need as much stuff being done at once and also have more (sometimes far more) dwarves, I tend to have two dwarves for each specialization job (metalwork/cooking/mason/etc), that way if one dies or is on break or whatever when something needs done, I often have a handful of workers do jobs that aren't as important to reach high levels in (either because high levels don't do much except increase speed, or because I just don't use the trade at my fort enough to warrant the training, including smelting metal/butchery/mechanics (that way lots of people can pull levers when needed)/etc) plus of course hauling and then a few dedicated administrators/hospital workers. All the rest go to my military. If a migrant shows up (or a mood happens) that has a higher skill in something they get subbed in somebody else gets subbed out.
Long answer, but figured there isn't really a "yes I specialize" or "No I dont" as it depends on how old my fort is.
-MB