By this I mean two 11 letter words that contain 22 different letters. Do words like these exist?
My answer is
No. There does not exist a pair of unique 11-letter English words that both each have 11 different letters and also do not share letters.
My method for finding this answer:
I grabbed all the 11-letter words from http://www.allscrabblewords.com/11-letter-words/ and made a list of words that contained 11 unique letters. This list has 363 words. I then checked each pair of words to see if any pair didn't share letters. Unfortunately, this check revealed 0 word pairs.
The OP asked a secondary question, and my answer is:
Question: Which 2 11-letter words contain the most distinct letters
It took less time than I expected to run through all combinations, but I found 15 pairs of words that have 20 distinct letters. The first pair that also each contain 11 distinct letters are: