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:
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.