Swords need sharp edges (duh). We have a more or less precise idea of how swords are supposed to be used from looking at historical manuals and it is obvious that brute strength and hammering is neither necessary nor the sensible thing to do. While the manuals do tell us to fight with all our strength, many techniques very obviously rely on having a sharp edge to do damage (for example push or pull slices). And of course, a regular cut or hau benefits greatly from a sharp edge as well, despite the impact trauma it would also inflict.
So that swords aren‘t clubs used to hammer at armor or break bones is pretty clear to anybody with a serious interest in the matter (what the „uneducated masses“ and Hollywood believe is an entirely different matter of course).
However, there is a lot of debate and discussion on just how sharp a sword should be. Terms such as „sword-sharp“, „paper-cutting sharp“, „shaving sharp“ and more get tossed around.
Let‘s first look at what decides how sharp a (sword) blade is. There are several factors at play here, I will look at mainly two of them: edge shape and polish. Steel alloy, hardness, etc all play a role as well of course. But that is a topic for another time 😉
When people say „the sharper a blade, the more fragile“, they mean the shape, the acuteness of the edge. The finer the angle, the less material there is at the edge to support it, the more prone it is to damage. The easier it cuts as well though… (all in general!)
Peter Johnsson gives a general edge angle for many medieval swords of between 20° and 25° (per side), giving an overall angle of 40-50°. This is not much different from the edges seen on many japanese nihonto and appears to be a good balance between acuteness and edge support. Now, edge shape is one thing, it doesn‘t say anything about the actual blade geometry behind that edge. The angle of the „primary bevel“ can vary immensely, as can the shape. A flat bevel that transitions in the convex edge is just one among many possibilities, other basic shapes include convex or concave/hollow ground bevels. It‘s a complex issue with many variables but there are certain basic principles that can help when trying to figure out why a sword cuts the way it does. Looking at swords that are generally seen as good cutters, it becomes obvious that they share several traits, such as thin, wide blades with a shallow primary edge angle (quite often the edge angle doesn‘t have to be particularly acute, a somehwat more obtuse edge can be required for durability reasons due to the thin blade body). A certain amount of stiffness is also quite beneficial though it is less crucial than on swords intended more for the thrust.
Now that we have (very briefly and nowhere near in depth as would be possible) looked at edge shape, let‘s move on to the other aspect that plays a role in edge sharpness: the polish. This is something that is still not well understood by many sword enthusiasts. An edge of a certain angle (say, 22° each side) can be either quite dull or very, very sharp. The basic shape of the edge (and, consequently, the durability!) is no different. The difference is in the polish. An edge that is taken up to 2000 grit and then stropped on leather bites substantially more aggressively than an edge of the same shape but left at, say, 600 or 800 grit. While the difference in performance isn‘t so noticeable on targets like plastic bottles or bare tatami mats, once the target is wrapped in a layer of linen (or several), the performance difference is immense. The less polished sword will quite likely just bounce off the target and only leave a mark due to its impact but it will not be able to actually cut the linen.
Seeing as naked people usually weren‘t what you‘d have encountered back then, I think this tells us something. I wouldn‘t go so far to assume that all medieval sword always were that sharp. Just as today one person‘s pocket knife is shaving sharp while another‘s rather dull, I would imagine it varied back then as well, for numerous reasons. A sword that has dulled a bit is still a useful weapon and I would imagine especially under campaign (or other) circumstances a quick touch up with a stone now and then would be possible but stropping with leather is a quite tedious process without a slack belt sander (though Matt Easton told that one anecdote about some tribe always polishing their sword on their leather shields during a rest (I think)).
However, I do believe that „in general“ we can assume that swords were honed to a level that might surprise many modern sword enthusiasts. I certainly think that it is time to abandon beliefs like „a sword only needs a blunt chisel-edge, it works with a splitting effect“ or „barely paper-cutting sharp is all that‘s necessary“. And especially „a very sharp edge (always) is very fragile and shouldn‘t be put on a sword“.