What you are asking for is very easily doable with JRiver Media Center.
You need to install JRiver MC on both laptops - you may do so in accordance with the JRiver licencing terms.
On the dedicated 'server' laptop, you should enable Media Server.
It's to be found under Tools>Options>Media Network.
Check the very first box: Use Media Network to share this library and enable DLNA.
An access code will be generated, and you will be given the option to activate authentication (add a username and password).
On the other 'client' laptop, you should also go to the Media Network options section. (It's to be found under Tools>Options>Media Network.)
Go to Client Options (when connected to a library server) by clicking on the little arrow next to the text.
Check the first three boxes.
The second one is a special case, and you can get help on setting that up properly another time, but for now it's ok to have it checked.
Under Audio Conversion:
Either check Don't Convert Audio, or Convert Audio if Necessary.
If you check the latter, you will be presented with a choice of which encoder you wish to use to convert non-streamable files to a streamable format.
Now on the 'client', you can go to:
File> Library> Library Manager
Click: Add Library
Click: Library Server (this will cause the client laptop to look for your server laptop)
Enter the access code (from the server laptop setup above), and authentication information when prompted. (This will allow the client laptop to connect to the server laptop and use the latter's library).
Now you can play anything on the client, and it will stream from the server laptop.
Altogether the above procedures make the music on the dedicated laptop available for sharing on the other laptop.
This should get you up and running. Ideally, you would also map the drive on the dedicated laptop that contains the audio files, and share it on the other laptop so that the path to the music is the same on both computers. Then there will never be a need to convert. If this seems like another language to you, you can ask for help or read about it elsewhere. UNC paths are another solution, but I know little about these to help.