Yes, because it will help you know if your mindsets align
As you have this discussion with your partner, you will get to know where you're both at as far as kids are concerned and whether you're on the same page.
The final decision will be yours of course but you need to address any concerns you both have especially if you're considering permanent methods.
No, you know your body best
Your partner should never force you into using a contraceptive you are not comfortable with, ever. Our bodies are different and what works for someone else might not be the best option for you.
The best person to help you choose what is right for you is your gyna in this case. They know details about your body and if you have any underlying conditions, it's important that you talk to a specialist instead of blindly making a decision out of pressure.
No, they might judge you for it
Couples aren't always on the same page when it comes to such issues. You might be okay with getting something like an IUD, because it's more effective, while your partner would rather rely on the pull-out or rhythm method due to certain beliefs.
This can cause so many problems in your relationship so maybe you should reevaluate whether you need to involve them in the process or not.