When I speak of somebody, e.g. Einstein, the German way of pronouncing this is Ein-sh-tein. Should I avoid German pronunciation in German names or not?
A couple things to consider:
Many well-known or famous names that are not native to English acquire a well-known English pronunciation that is not the same as in the native language.
English-speakers unaware of the proper pronunciation could fail to recognize it, so care is needed.
Even though Einstein did not emigrate to the US until 1933, and was an American citizen for just the last 15 years of his life, this is not immediate "street" knowledge. In my opinion the popular view is likely that he's "considered American" and it would be typically expected that his name should be pronounced "American." Your audience could wildly differ here with what they would know about him by default, the point is to take it into consideration.
If you are educating someone about Einstein, with you being German and natively knowing the language, you are certainly positioned to educate them about the proper native pronunciation (as well as many other things about Einstein), and definitely should do that.
There was a good example of this in the recent movie "Spy". A character says "I'm going to Capri" (pronouncing the name with a long, flat 'a' as in 'car') rather than the more normal British pronunciation where the 'a' is very short. The other characters pick up on this, and it serves to underline their dislike of that person.
Typically, an attempt to force an original-language pronunciation in normal English conversation may a) 'sound funny', thus drawing attention; b) appear pedantic, thus making that attention negative; c) appear to be attempting to assert superiority (e.g. of knowledge), also resulting in a strong negative response
To use an extreme example, consider saying "My trip starts in Par-ee, then heads south, taking in Meunchen and ending in Roma". While one can argue that the name used by a city's own inhabitants must be more correct, it would be extremely odd not to use the Anglicized versions when speaking English.
They are effectively translations: in some cases this becomes clear because the word itself has changed. In the Ein-s-tein/Ein-sh-tein case, I would still think of the first as "the English translation of" the original even though the written word has not diverged.