Either use 'append' with >> or use braces to encompass the I/O redirections, or (occasionally) use exec:
ssh host tail -f /some/file | awk ..... > /some/file &
ssh host tail -f /some/file | grep .... >> /some/file &
or:
{
ssh host tail -f /some/file | awk ..... &
ssh host tail -f /some/file | grep .... &
} > /some/file
or:
exec > /some/file
ssh host tail -f /some/file | awk ..... &
ssh host tail -f /some/file | grep .... &
After the exec, the standard output of the script as a whole goes to /some/file. I seldom use this technique; I usually use the { ...; } technique instead.
Note: You do have to be careful with the braces notation. What I showed will work. Trying to flatten it onto one line requires you to treat the { as if it were a command (followed by a space, for example) and also to treat the } as if it were a command. You must have a command terminator before the } — I used a newline, but an & for background or ; would work too.
Thus:
{ command1; command2; } >/some/file
{ command1 & command2 & } >/some/file
I also have not addressed the issue of why you have two separate tail -f operations running on a single remote file and why you are not using awk power as a super-grep to handle it all in one — I've only addressed the surface question of how to redirect the I/O of the two commands to one file.