Is there a way to sharpen stone weapons?
In a society with no metals whatsoever, which is therefore using stone for their weapons, can the cutting edges of those weapons be sharpened?
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Historically (and prehistorically), most sharp stone tools (projectile points, scrapers, knives, etc) were made from obsidian (volcanic glass which lacks a crystalline structure), cryptocrystalline silicates (cherts, such as flint), or fine grained volcanic rock (e.g. fine-grained andesite).
Generally speaking, when one wants to make a new tool, the maker will use the percussive force of a stone against a "core" of toolstone to knock a roughly tool-sized flake off of the core. The flake is then be shaped and sharpened via percussive flaking (hitting the tool blank with a rock or antler to knock off smaller flakes), and is ultimately sharpened using pressure flaking (pressing the tip of an antler or dense stick into the tool to knock off small flakes). The flaked edge can be incredibly sharp (e.g. obsidian flakes to blade whose cutting part can often be measured in atoms of thickness; ages ago, when I was actively learning to make stone tools, I sometimes shaved with obsidian flakes). This general process is called flintknapping.
As a knapped tool is used, the edge will wear (as flakes are knocked off in a non-systematic fashion). The edge can be refreshed by using the same knapping technique as is used to construct the tool in the first place.
A great reference for the practice of flintknapping is Dr John Whittaker's book Flintknapping: Making and Understanding Stone Tools (disclosure: John is a good family friend; the book was illustrated by my mother—that said, it is an authoritative text in the field).
That being said, it is likely that stone weapons will come in one of two general varieties: projectiles (like atlatl darts or arrows), and "bladed" weapons with stone blades embedded into a more durable medium (such as the Mesoamerican macuahuitl). In either case, using the weapon is likely to either leave the stone parts relatively unaffected, or to completely break them. It is unlikely that one would seek to sharpen the edges of such a weapon, and more likely that one would simply replace the stone parts (i.e. replace the obsidian blades of a macuahuitl). Stone itself is generally quite brittle, hence one would not expect to make a weapon entirely out of stone (e.g. a stone sword would likely be quite useless, Minecraft notwithstanding).
 It is worth noting that stone tools can be quite useful, and that, for example, it is possible to buy obsidian surgical tools.
When a flint or other stone blade becomes dull, it is sharpened by either gently chipping flakes off the edge (as was done in final shaping for shaped blades) or sometimes by rubbing the edge on another stone to wear away the dulled "points" of the natural serrations in a shaped edge and expose the unworn "throats" of the individual chips that formed the edge.
The only stone tools that can't be sharpened to original capability are flake blades -- and these are quite primitive types that are usually replaced (as a culture's knapping tradition evolves) by shaped tools that can be hafted and last better.
You can sharpen stone tools with sharpening stones.
I like this Primitive Technology series. Here Mr. Plant makes a basalt axe head then sharpens it with progressively finer stones. It is much the same as one would do with a metal tool. In the same video he makes a chisel from mudstone (which I would not think would make a very good chisel) and proceeds to cut down a tree with it. The chisel is sharpened with grinding on other stones, just like the axe.
Never heard of obsidian blades?
Obsidian is a confusing element. It cannot be considered a mineral, because as a glass it is not crystalline, whereas minerals are. But at the same time, it is classified as a mineraloid, since it has mineral qualities, but they are too variable to be pure mineral. Get it? Me neither.
First Encounters The first-ever known historical use takes us into the Acheulean age. This age is classified through the first practices of stone tool usage. Obsidian blades were often found with the remains of Homo erectus, and it is believed that these technologies developed nearly 1.76 million years ago.
The way that obsidian fractures are so sharp, that it wouldn’t even require the tool to be sharpened into anything.
Or, similarly, never heard of flint?
Flint breaks and chips into sharp-edged pieces, making it useful for knife blades and other cutting tools. The use of flint to make stone tools dates back hundreds of thousands of years, and flint's extreme durability has made it possible to accurately date its use over this time. Flint is one of the primary materials used to define the Stone Age.
If there is still enough body remaining, you can simply chip it some more to sharpen the edge.