Here is their FAQ section:
https://www.whatsapp.com/faq/
How do I use WhatsApp on my computer?
To provide you with a richer communication experience, WhatsApp is now accessible both on your phone and your computer. WhatsApp desktop app and WhatsApp Web are computer based extensions of the WhatsApp account on your phone. The messages you send and receive are fully synced between your phone and your computer, and you can see all messages on both devices. Any action you take on the phone will apply to WhatsApp on your computer and vice versa.
So my guess is that aren't stored on their server.
If you're happy and you know it......it's your meds.