As so often, the response needs to be "compared to what"?
If it's true that $E[Y|X=x]=x\beta$ then OLS estimates $\beta$, regardless of the correlation, but that $\mathrm{var}[\hat\beta]$ is affected by the correlation. Suppose, to start with, that you do want to estimate $\beta$.
Now, it's clearly not true that "if your coefficients are statistically significant, you can totally ignore the VIF value and cross off multicollinearity as of no concern". This would be true if statistical significance, in some binary yes/no sense, was the only reason you cared about precision. It isn't -- higher precision is important because you want to estimate $\beta$, and so you want $\hat\beta$ to be close to $\beta$. So person B is completely wrong in that sense.
On the other hand, what are you going to do? If there's high correlation between the $X$s and you do want to estimate $\beta$, you can't actually do much about it. If you leave a variable out of the model, to get a design matrix $Z$ with less correlation between columns, you'll have a model
$$E[Y|Z=z]=z\gamma$$
and $\gamma$ will be estimated more precisely than $\beta$ was. But $\gamma$ will be different from $\beta$, and there's no reason to expect $\hat\gamma-\beta$ to be smaller than $\hat\beta-\beta$. You'll (typically) get a worse estimate of $\beta$ by changing the model. You do need to think carefully about whether $\beta$ or $\gamma$ is what you want to estimate, but that decision doesn't depend on the VIF.
If you're doing prediction then you still care that your predictions will be less precise than they would have been with lower correlation between the $X$s. Again, though, what are you going to do? If your best predictive model includes two correlated variables then it includes two correlated variables and it's the best predictive model, so ex hypothesi removing one of them will make the predictions worse.
It might be the case that you could collect supplementary data. In that case collinearity matters a lot. You'd benefit much more from collecting data that expanded $X$ in a direction where it was thin, decreasing the correlation.
