How to participatein heated loud discussions as an introvert?

Don't force yourself, you'll be uncomfortable and you'll likely feel embarrassed later. You don't have to talk a lot or loud in order to participate and be respected. What matters is if people listen to you when you do talk. If you're with the right people, they will.

[i]Speak your truth quietly and clearly[/i] ~ Desiderata

Do you really want to be part of it... They'll always do it better than you... That's not a reflection on you, but on them... Sometimes talking softly can work as they may shut up to listen.... But no guarantees

As introvert we need to be interested in the subject before we make the step to become involved. If the subject isn't interesting to us nothing will make us join in.

It doesn't need to be a bar, for example, we just had a christmas party at work, where everyone was loudly joking around and I found myself to be too introverted to join in.

I don't engage in discussions that are loud and heated, folks who are like that aren't capable of being objective and not really very intelligent.