I have a p1 and from my experience the buttons require a lot of pressure because that's just the way it's made. If you want to you can open it up and take a look inside. (I would only open it if the buttons were rarely/not responsive). Be careful if you choose to open it. Slowly make your way to the buttons and clean them out with q-tip/see if anything is preventing them from working. I hope you have fun with your new p1!
I have the same problem with my P2. You probably need to clean the interior like Tamaninjacat said. If you don't want to open your Tamagotchi, try using a thin object to push the buttons like a pen (one that has run out of ink so that it doesn't get ink on the buttons).
This tends to happen a lot with the older vintage ones. It's like the buttons decrease over time. Was it bought NIP or previously used? I've found the the NIP buttons tend to be 'harder' and easier to use.