My thinking is that two handers are not weapons of convenience or of self-defence. They were to be used in battles that you knew were coming up, and you just had the sword out and ready.
If you were marching from point A to point B, it was stored away. For defending yourself along that journey from surprise attackers you'd most likely have a dagger or shorter sword about your person.
I agree, I've got a short history of
Galloglass's - Gaelic mercenary warriors - here at home (I love dipping into those Osprey books!) and they fought battles with two-handed weapons, although because they have nordic origins it tended to be double headed axes. But they also carried side arms, ranging from dirks, hunting bows to scabbarded broadswords - as TomG points out for immediate use if required. The two-hander would be brought up when a battle was about to start.
For that purpose, although the book is not 100% clear, it seems to be more or less given that each galloglass had at least one attendent/knave. And it would be the attendents responsiblity to store (and sheaf if it had a sheaf!) and then bring it to his master when required, I guess.
I don't know how expensive the weapons themselves are - compared to other smaller versions, but the actual soldier who would be wielding it
was expensive to train and to use these big weapons well (He'd have to big and strong and so be built up from childhood), hence these were higher status troops. And so I'd expect he'd have a retinue to service his needs, as he was a more valuable asset.
You see them same thing with knights - they would have gone into battle with a retinue - who would look after his armour, bring him new horses, etc...Obviously the richer you were the bigger the retinue you could bring.