The procedure looks like this:
- find a seller (for example at olx website, close to your location), who owns a scooter you like,
- contact with him (via email or phone), try negotiate the price,
- make a payment for the scooter (best in person via cash)
- enjoy your purchase
To buy, no need for any special documents, just money.
To drive - you are required to have a driver licence - one of "A" category, or "B" (when you have it 3 years or more in Poland)