Assume you either do not have ssh installed, or cannot do do not want to share keys/secrets.
Suppose user1 is in group1, and user2 is in group2, and both user1 != user2 and group1 != group2.
Create a shared group, group3s.
addgrp group3
Add both user1 and user2 to group3s.
Create a directory in a mutually accessible place, where one user owns, but has group ownership of group3s.
#as user1,
mkdir $place/shared && chown user1.group3s $place/shared && chmod 770 $place/shared;
#as user1 or user2,
cp $file $place/shared && chgrp $place/shared/$file && chmod 660 $place/shared/$file
But, suppose you cannot create the new, shared group, and place both users in that group?
Create the directory and give it 770 permissions,
mkdir $place/shared && chown user1.group1 $place/shared && chmod 770 $place/shared;
Then, as root/admin, change the group ownership to the other user's group,
sudo bash
chgrp group2 $place/shared && chmod g+s $place/shared
The command chmod g+s sets the setgid bit so that files placed into the directory have the group ownership set to group2.