I created both a DSA and an RSA key for communication between two SSH servers. Each server has the same DSA and RSA key in their .ssh folders. Yet when copying files from one server to another using a client terminal, I get a password prompt. Why is that? What should I do to bypass the password prompt on each instance of scp?

How did you do that?

Lets say you did the job correctly,
Possible issues:
home directory must be 755!
I lost 2 days to find that...
same directory .ssh must be 700...

Creating the key:
The following command was run on both servers and on the local client in separate terminals.

ssh-keygen -t dsa

I pressed enter three times to create the files .ssh/id_dsa and .ssh/id_dsa.pub. I then performed the following command on both servers.

scp .ssh/id_dsa.pub [user@hostname]:.

I entered my password for the scp command and copied the key from each server onto the other.

cat id_dsa.pub >> .ssh/authorized_keys

This installed the key on the servers.

Next, I copied the key from my local client onto both servers and installed them in the same way. Finally, I performed the following commands for each server on the local client:

scp [user@hostname]:.ssh/id_dsa.pub . ; cat id_dsa.pub >> authorized_keys

All machines are authorized to view and copy files on one another. The local client is able to share files between both servers with no password input. One server is able to copy files from the other with no password input. However, the second server is unable to copy files from the first without password input. And when I do enter the password, it breaks the operation.

EDIT: I'm a complete newbie. What do you mean by "must be 755" or "must be 700"?

File permission:

ant:/home/vbe $ ls -ld .
drwxr-xr-x  55 vbe        bin          30720 Jun 25 15:33 .
ant:/home/vbe $ ls -ld .ssh
drwx------   2 vbe        bin           1024 Jun 15 16:47 .ssh

Pretend your login directory is: /home/snorton

Then

chmod 755 /home/snorton
cd /home/snorton
chmod 700 .ssh

If you got the idea, I wonder still if you did not overwrite one...
Start all again, but this time lets say you have local, server1 server2...
once keys are generated, copy your local fresh one for safety:

cp -p id_dsa.pub id_dsa.pub.mylocal