Very satisfactory saw vises are quite easy to make using inexpensive or scrap material. In an hour or so you can make one like Lie-Nielsen uses. I made mine from a 2x4 and some plywood offcuts:
Although Lie-Nielsen uses scrap pieces of leather for "hinges," I used standard hinges because they keep the jaws even. Originally the jaws were rectangular but I found them to be knuckle bangers when filing, so I remade them as you see here. This design is made to be clamped in a shoulder vise and you can make it long enough that you don't have to reposition the saw during filing. It's a bit cumbersome to position the saw while you tighten the vise but, all in all, it works very well.
Since I still don't have a vise on my new workbench and I was feeling inventive one day, I decided to come up with a free-standing design using a rule I enjoy: I restrict myself to scrap materials already in my shop. It sits on top of the bench and is held in place with a clamp or holdfast:
The vise is made from Douglas-fir 2x4 offcuts and is 26 inches long to accommodate my longest handsaw. When I want to file from the other side, I leave the saw clamped in place and just turn the vise around. Oversize holes in the legs for the lag bolts allow the jaws to move just enough to clamp the saw in place securely. The legs are in shallow dados and are screwed in place from the inside. One thing I learned is that it helps to plane the jaws with a slight toe-in so they come together at the top first as a way of minimizing vibration when you are filing. I used a french curve to approximate the cut-out for the saw handle. The vise works very well. The only thing I don't like about it is that the clamp handles stick out on one side of the vise, but they don't really impede filing.
One thing you may notice is that the jaws are deep enough that you can't file backsaws with narrow blades. I liked the design well enough that I made another one for backsaws:
The design is very similar, smaller obviously, except that I planed a rabbet in the inside bottom of each jaw large enough that the back of the saw can fit. That way, I can file a backsaw with a blade as narrow as an inch. This keeps the jaw very stiff while still allowing sufficient clearance. You could do the same thing with the Lie-Nielsen design.
I am happy with my design, although I have seen much fancier ones made by other woodworkers. One thing I like about it is that the saw is raised up to just the height I like for filing standing up, which I prefer.
I hope I have gotten you thinking about designing one for yourself.