This question is motivated by this one. I looked up two sources and this is what I found.
A. van der Vaart, Assymptotic Statistics:
It is rarely possible to compute a profile likelihood explicitly, but its numerical evaluation is often feasible. Then the profile likelihood may serve to reduce the dimension of the likelihood function. Profile likelihood functions are often used in the same way as (ordinary) likelihood functions of parametric models. Apart from taking their points of maximum as estimators $\hat\theta$, the second derivative at $\hat\theta$ is used as an estimate of minus the inverse of the asymptotic covariance matrix of e. Recent research appears to validate this practice.
J. Wooldridge, Econometric Analysis of Cross Section and Panel Data (the same in both editions):
As a device for studying asymptotic properties, the concentrated objective function is of limited value because $g(W,\beta)$ generally depends on all of $W$, in which case the objective function cannot be written as the sum of independent, identically distributed summands. One setting where equation (12.89) is a sum of i.i.d. functions occurs when we concentrate out individual-specific effects from certain nonlinear panel data models. In addition, the concentrated objective function can be useful for establishing the equivalence of seemingly different estimation approaches.
Wooldridge discusses the problem in wider context of M-estimators, so it applies to maximum likelihood estimators also.
So we get two different answers for the same question. The devil in my opinion is in the details. For some models we can use hessian of profile likelihood safely for some models not. Are there any general results which give conditions when can we do that (or cannot)?