I only ask because I would be buying the full version mainly for servers and I've heard that playing on servers is harder on your computer than singleplayer. I don't have a great laptop for gaming and the demo was okay frame rate wise (20-30 fps) I just don't want to go out and buy the game to find that I'll only get 10-15 fps online haha.
How to test minecraft multiplayer?
You could test it if you had a copy of minecraft that wasn't legal, but other than that there's no legal way to test online multiplayer without buying the game.
I play on servers a lot and even have all of my friends join a party and we do Hunger Games, and other mini games online. My computer isn't great either (it's really cheap with 4GB RAM And an Intel Celeron processor), but i still get around 30 FPS. The performance you have on single player will approximately be the same as it is on a server, provided that you have a strong and fast INTERNET CONNECTION. The internet connection is really what matters more than the processing power when playing online. The processing power of the computer does come in to play, sometimes I will lose a one-to-one fight with another player when we both have stone swords, but most of the time it can be overlooked. As long as you can play minecraft smoothly on your computer and have a good internet connection, your performance will be fine on servers.
You can also increase the performance of minecraft if you change the game settings to be simpler, such as Fast Graphics instead of Fancy Graphics, reduce smooth lighting, etc, (anything found in the Video Settings).
ALSO, you could test minecraft on your computer if you had a friend who has purchased minecraft. Simply download the minecraft launcher for your computer from minecraft.net, then sign in with your friend's account and test the gameplay on a server.
The one I use is mc.hypixel.net
As long as your computer can run Minecraft smoothly, I'd recommend buying the full version because you won't regret the purchase (you'll only lose around $27 for a game that truly I play more than any $60 console video game), and you'll have something you can play anywhere on a laptop.
Have fun and maybe we'll meet in the Hunger Games.