How to 'grep' a continuous stream?

I think that your problem is that grep uses some output buffering. Try

tail -f file | stdbuf -o0 grep my_pattern

it will set output buffering mode of grep to unbuffered.

Turn on grep's line buffering mode when using BSD grep (FreeBSD, Mac OS X etc.)

tail -f file | grep --line-buffered my_pattern

It looks like a while ago --line-buffered didn't matter for GNU grep (used on pretty much any Linux) as it flushed by default (YMMV for other Unix-likes such as SmartOS, AIX or QNX). However, as of November 2020, --line-buffered is needed (at least with GNU grep 3.5 in openSUSE, but it seems generally needed based on comments below).

I use the tail -f <file> | grep <pattern> all the time.

It will wait till grep flushes, not till it finishes (I'm using Ubuntu).

If you want to find matches in the entire file (not just the tail), and you want it to sit and wait for any new matches, this works nicely:

tail -c +0 -f <file> | grep --line-buffered <pattern>

The -c +0 flag says that the output should start 0 bytes (-c) from the beginning (+) of the file.