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Do we have a reasonable idea of how well we've mapped our local neighborhood of stars? That is, about what fraction we know about vs. what fraction are unseen (too dim, too much dust, not enough observing time, etc.).

Something like, "We're almost certain to have found all stars out to __ pc, we expect to have found > 90% [or whatever percentage is convenient] out to __ pc, we think we've found only -% in the Milky Way" would be perfect, especially with a reference. If I could have just one distance my current interest is ~2000 pc but I'll take whatever can be found. (Clearly we have at least fair coverage of ~60 pc to map the Local Bubble.)

Charles
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One way for you to find out would be to look at lists of stars within varius distance limits of Earth.

For example Wikipedia has:

https://en.wikipedia.org/wiki/List_of_nearest_stars_and_brown_dwarfs

That lists all the stars and brown dwarfs out to a distance of 5.0 parsecs or 16.3 light years from earth.

Counting by star systems and not by individual stars, there are 7, counting the solar system out to about 2.5 parsecs or 8.15 light years. A sphere with a radius of 5 parsecs would have 8 times the volume, so there should be about 8times as many star ssytems, or about 56, out to 5 parsecs.

I counted 56 star systems out to about 5 parsecs. So the percentage of stars whhich have been discovered does not seem to diminish with distanc eover that samll distance.

So you should search for lists of stars within various distances of the Sun. And you should calculate the relative volumes of the distances in the various lists. And then count the numbers of stars within each volume to see how they compare to the relative volumes of space.

Finding lists that number the stars would be very convenient for you.

And I suppose that some such lists may discuss how complete they are.

Added on Sept. 10, 2021.

If the relative numbers of stars within two different distances of Earth - for example 10 parsecs or 20 parsecs, are proportional to the relative volumes of those two spheres, you will know that the samepercentage of stars has been discovered in both of the spheres, and the discovery rates are identical.

But if the number of stars in the larger sphere of space is smaller than would be proportional to the relative volume, you can deduce that the precentage of stars shwich have beendisocered is maller in the alrger sphere, and out knowledge of the stars is startomg tp be less complete within this distacne range.

M. A. Golding
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    I can do that part easily, but I'm asking about completeness here. How far out do we know (with a reasonable degree of assurance) that our catalogs are complete? – Charles Sep 09 '21 at 19:34
  • Probably about 10 parsecs; see https://astronomy.stackexchange.com/a/45198/7411 – Peter Erwin Sep 09 '21 at 20:08
  • @Charles On 09-10-2021 I added to my answer a few paragraphs explaining the obvious way that comparing the relative volumes and numbers of stars in two sun-centered spheres with different diamters can show the relative completness of discovery within the two spheres. I did not think that someone smart enough to ask questions here would need a detailed explanation. – M. A. Golding Sep 10 '21 at 17:24
  • This is answered more accurately in the near duplicate, but there is space to answer the non-duplicate part of this question, which isn't yet addressed. – ProfRob Oct 11 '21 at 06:44
  • "The true stellar density near the Sun is estimated as 0.004 stars per cubic light year, or 0.14 stars per cubic parsec.". I suppose that at greater distances we would see a smaller proportion of dim stars. – Keith McClary Oct 11 '21 at 17:22
  • The math on this answer works great, to a point. After (I forget whether it's 100ly or 100pc or maybe even an order of magnitude more), you run into the problem of the Milky Way being a disk, and then you have to recalculate based on diminishing returns of a sphere sampled within a disc... – GrasshopperTaji Dec 21 '21 at 21:20
  • @GrasshopperTaji The thickness is about 1000 ly, so by the time you're at 500 ly you have to take it into account. (We're almost exactly on the galactic plane, maybe ~50 ly above.) At a few kly a cylinder is a better approximation. At 17 kly you hit the edge of the galaxy and have to take that into account, and at 22 kly you need to take into account the galactic bulge as well. – Charles Dec 09 '22 at 20:52