I have a concrete hole that was drilled a little bit bigger and the lag shield anchor is a bit loose. How do you tighten it? Insert paper or wood?
Shaving some slivers of wood to take up the slack will work OK today, you probably won't achieve the rated holding power, and before long the wood will deteriorate and the plug will come loose. So don't do that, especially if it's something critical where a failure could do damage or injure someone.
You could use an epoxy that is rated for metal and concrete, if the hole is cleaned out well, that should hold pretty well, but you can never be sure, so again that's no good for a critical application.
The easiest thing to do, if it will work, is fill the hole and move over a few inches.
If it's block or another hollow material, you could use a toggle bolt.
If it will work for you, can drill the hole larger and use the next size up lag shield and lag screw.
If you're already using the largest size lag shield available, you may be able to switch to an epoxy anchor, but that will be expensive and you have to do things exactly right for those to hold.