In my opinion, the only time accent becomes important is when it is so strong that it makes it difficult for the listener to understand the person speaking. If you know you have a strong accent and/or want to retain your accent, then you will probably need to accept that some listeners might ask you to repeat what you have said. In fact, this happens across different regions of the UK, where English is spoken.

This can be a controversial topic, because sometimes people feel that their accent is part of their identity, and because they are proud of their identity they don't want to have to do anything to try to reduce their accent. However, I think that two people who want to have a conversation BOTH need to adapt if they want to communicate with each other. Maybe a native speaker might need to talk slower if the other person is not a native? I think we would all agree that this is reasonable and not in any way offensive to the native speaker? There are native speakers of English that other native speakers find very difficult to understand because of their accent (and also sometimes because of dialect or sociolect). In that case if the people are good communicators they will adapt in some way, and maybe that will include switching to a more neutral accent. And I don't think that the person with accent should be offended. I believe it's the same with Italian (which I am learning), but I can't really identify different Italian accents. I imagine it is similar with Spanish? In the UK, we used to only hear one particular type of accent on TV and radio, and non native students used to want to learn that accent. Now we are very familiar with a wide range of accents on TV and radio, and we generally don't insist that our students must study a particular accent.

I don’t think the accent is important when you communicate with others.