What is the quickest way of replacing 0 by 1 and vice-versa in a stream?

You can use tr for this, its main purpose is character translation:

echo 111111100000000000000 | tr 01 10

Your sed command replaces all 0s with 1s, resulting in a string containing only 1s (the original 1s and all the replaced 0s), and then replaces all 1s with 0s, resulting in a string containing only 0s.

On long streams, tr is faster than sed; for a 100MiB file:

$ time tr 10 01 < bigfileof01s > /dev/null
tr 10 01 < bigfileof01s > /dev/null  0.07s user 0.03s system 98% cpu 0.100 total

$ time sed y/10/01/ < bigfileof01s > /dev/null
sed y/10/01/ < bigfileof01s > /dev/null  3.91s user 0.11s system 99% cpu 4.036 total

Although tr is the right tool for this job you can do it in sed using the y (transliteration) command rather than the s (substitution) command:

$ echo '111111100000000000000' | sed 'y/01/10/'
000000011111111111111

y is basically sed's internal implementation of tr - with all the overhead that implies.

A way is echo "111111100000000000000" | sed 's/1/2/g;s/0/1/g;s/2/0/g'