try

 cd /path/1
 find . -type d -print | sort > list1.dir
 find . -type f -print | sort > list1.file
 cd /path/2
 find . -type d -print | sort > list2.dir
 find . -type f -print | sort > list2.file

• sort is used to ensure same order, and a smaller result for diff or comm
• you might use absolute destination file name, so that list1.file and list2.file will not "polute" results.

Note that directories on Unix are just one of many types of files. With find, you can search for them with -type d, or use the / qualifier in zsh globs. Other types of files include regular files (-type f, . glob qualifier, maybe what you meant by file), but also symlinks (-type l / @), devices, fifos, sockets…

To get the files of type directory, you can do:

find dir1/ -type d

And for files of any other type:

find dir1/ ! -type d

And same for dir2.

Now that comes with 3 main problems:

• the printed paths will start with dir1/ for dir1 and dir2/ for dir2 which would make the comparison more difficult.
• the order will be random.
• the file paths are written one per line, but the newline character is as valid as any in a file path, or in other words, file paths can be made of several lines, so the output is not post-processable reliably.

Those can be addressed with GNU find and sort by using:

find dir1/ -type f -printf '%P' | LC_ALL=C sort -z

Where:

• %P prints the path of the file relative to dir1
• we sort the list (in the C locale as file paths don’t have to be made of text)
• we use NUL-delimited records instead of lines as 0 is the only byte that cannot occur in a file path.

Now, you can compare the list with:

list() {
  find "$@" -printf '%P' | LC_ALL=C sort -z
}
echo Directory differences:
comm -z3 <(list dir1/ -type d) <(list dir2/ -type d) | tr '' 'n'
echo Non-directory differences:
comm -z3 <(list dir1/ ! -type d) <(list dir2/ ! -type d) | tr '' 'n'

That output is not post-processable reliably as we translate the NULs back to newline for displays an comm uses TABs to separate the columns which again is valid in a file path.

Alternatively, you can get the lists in zsh arrays and use its array comparison operators:

dirs_in_dir1=( dir1/**/*(ND/:s:dir1/::) )
dirs_in_dir2=( dir2/**/*(ND/:s:dir2/::) )
nondirs_in_dir1=( dir1/**/*(ND^/:s:dir1/::) )
nondirs_in_dir2=( dir2/**/*(ND^/:s:dir2/::) )

Then:

dirs_only_in_dir1=( ${dirs_in_dir1:|dirs_in_dir2} )
dirs_only_in_dir2=( ${dirs_in_dir2:|dirs_in_dir1} )
nondirs_only_in_dir1=( ${nondirs_in_dir1:|nondirs_in_dir2} )
nondirs_only_in_dir2=( ${nondirs_in_dir2:|nondirs_in_dir1} )

And do what you have to do with those arrays, like print them raw on 1 Column with:

print -rC1 -- $array

(or NUL-delimited so it can be post-processed by adding the -N option).