First, let me say I do this as a hobby, on my own stuff, so don't have a lot of experience.
Situation: A Shilen SS bbl, short chambered for .22-250. Clymer solid pilot reamer. (Both from Brownells)
It appears to me the reamer pilot is a very tight fit in the bore. Problem is, I don't know how tight is "too tight". Looking for some guidance.

What are you using for lubrication? Is the pilot to tight you have to push the reamer in before it seat's? The pilot is only for guidance and should spin in the bore with little resistance.
Try putting the reamer in the chamber until it stops, and the back it out a 1/8" or so and see if you can turn it with your fingers. Make sure that you alway's turn the reamer in the cutting direction. You should be able to turn the reamer with your fingers with only the pilot touching the inside of the bore
Check for chips making the pilot bind in the bore.

Well, experience will tell you a lot, just keep on chambering and learning!! My first chamber job was interesting.

You need reamers with removable pilot bushings.

I have bushing sets for all bore sizes, in 0.0002" (that is 2 tenths of a thou) increments for the precision stuff and up to 375. Then in 0.0005" increments for the big bores. Got several hundrred bushings in the tool chests. You do not need this many, but over the years the collection builds up. As the bore size goes down, like 22 and 6mm, the need for accurate bushing fitting becomes more important. Same for match rifles, if I am building a 1000 yard 30 cal., I fit a very snug bushing. Big bores like 416 or 458 are less critical, as the rifle will not be used for shooting sub 1/4" groups.

Bore sizes are all over the place, even for the same barrel manufacturer. So, gotta fit a snug bushing to the bore. I have rods (about 8 inches long) made up that take a bushing on the end, and have a retainer screw. Carefully push the bushing into the bore, past the point where it will wind up when on the chamber reamer, and keep trying different sizes until I have resistance. Sort of like learning how to use a feeler gauge when adjusting valves on engies with solid lifters, takes a few times to develop the touch.

Bore must be clean and slightly oiled when trying the bushings for size. And be sure the centers in the barrel do not have a burr, you usually have to recut them, as the centers are damaged when the barrel maker does the final contour of the barrel.

A trend in big bore barrels from several makers is to increase the bore size. Example: 416 barrels should have a .408 bore. Lately I have seen .410 and .411 bores in 416 barrels. Stick a reamer with a solid pilot at about 0.407" into a .411 bore and the reamer rattles around. So, gotta have bushings in a good range of sizes to fit to the bore.

And, for kicks, I have long rods made up so I can push a bushing completly through a barrel, like using a pin gauge to measure the bore inconsistencies. You will be suprised how much the bore diameter varies in those barrels we pay a lot of money for.

Search the archives, lots of good information here on chambering, reamers, etc.

Some hints:

If you plan on doing much chambering, you need a muzzle flush coolant system. Gives a better finish to the chamber, lets you run at higher speed, and the reamers last longer.

You need a good set of indicating rods for aligning the bore centerline to the lathe axis.

And a high quality floating reamer holder is a must.

JGS and Pacific Tool bushings are interchangable. Clymers are not.

JGS and Pacific Tool use a retainer screw for the bushing. Clymer uses a snap ring. The screw is much easier to use.

Pacific Tool extends the reamer flutes under the pilot bushing and out to the screw. Better for coolant flow when using a muzzle flush system. I notch the screw head, in alignment with the flutes, greatly improves coolant flow.

I have not bought a Clymer reamer in over 15 years. I will let you form your own opinion about this statement.

Your solid pilot reamer should slide in with very little "shake". Clearance of .0005 or less is ideal. If you push the reamer in and it is hard to extract with your thumb and forefinger, it is too tight. You can set the reamer up in the lathe a hone lightly with a fine ceramic hone. You will probably not have to remove more than a couple of ten thousandths. Spin the reamer fast when doing this and keep the hone moving.
As John mentioned, it is probably preferable to use floating pilot reamers and to have a selection of pilots in .0002 increments. That isn't the way things always are though and it is necessary to do well with what we have. I prefer floating pilots too but don't turn my nose up at solid pilot reamers either! They have been working very well for many, many, years. Regards, Bill.

Thanks for the replies folks. I think I will take Mr. Leeper's suggestion and do a bit of honing. I had thought about that, but wasn't sure how to procede...