It depends on the individual.
I know older German women who were war brides, came over here in the late 50’s and early 60’s. Many of them have had successful careers and though they speak English well they still have a heavy accent and speak with some non-standard English phrases.
Others immigrants I have met speak English with no accent at all but if you read something they write it is clear that English is not their first language.
I don’t think it’s that big of an impediment to have an accent. We have many non-American scientists, engineers, doctors etc. working in this country at highly technical jobs who still speak with accents from their country of origin.
Yes, you do need to be able to communicate but I don’t think you need absolutely perfect English diction to be able to do that. It certainly helps but there are other factors as well such as one’s competence in a given field.
For example we had many German and Polish scientists who immigrated to the US after the war who never spoke perfect English but who nevertheless did valuable work.
I have worked with many people who have strong foreign accents and it doesn’t seem to keep them from getting hired.
Where it does make a big difference is in written language. To be able to write perfectly in a second (or 3rd, 4th) language is very difficult. Very few can master that. But then many people who are native English speakers cannot write well in English.
Less than perfect English skills may not be an impediment to someone who already has professional credentials before emigrating to the US.
That is not the case if you are bright and hardworking, but without those credentials.
Perhaps persons in technical fields where human communication isn’t important can escape this general rule. But they are the exception.