"Is" sounds better to me. I parse it as
"being is necessary," that is to say
"being persuasive-and-assertive is necessary..."
It is possible to hear it "being persuasive and assertive" as an abbreviated form of "being persuasive and being assertive," so that you mentally supply the second "being" and the sentence becomes
"being and being are necessary..."
If I were faced with the problem of writing this sentence in real life, I'd sigh and rewrite it so as to sidestep the problem:
In this job, it is necessary to be persuasive and assertive.