Serial connection between 2 computers
Solution 1:
First, when connecting two computers back-to-back via serial RS-232, you need to use a null-modem connector (crossover) so that you connect input to output and output to input. You can get a null-modem cable, or you can get an adapter to replace the gender changer.
Once connected, your thought of using putty is on the right track. You need to be sure both ends are trying to talk with the same baud rate (bit transfer speed), and using the same parity (even, odd, or none). Both ends will probably default to the same, but you should be sure to check them.
When you get these tested and are comfortable with the setup, you can try enabling the getty on one end (getty is the program that prompts for "login:"). You should be able to see the login prompt in the putty session on the other end.
Be sure to only enable getty on your headless servers, not on the computer on other end of the serial connection - they will both think the other's "login:" prompt is the login name and start echoing back and forth.