Hugues,
This should be relatively straightforward given one very crucial assumption, that we will get to. Let's establish some notation. Let's define $X$ to be the random variable obtained by randomly selecting a data point from your set and classifying it using your classifier. $Y$ as the random variable obtained by randomly selecting a data point from your set and getting it's gold standard class label. And $Z$ as the random variable obtained by randomly selecting a data point from your set and getting its true label.
Now let's summarize the information we have so far. The things we know or believe we know are
$$
P(X|Y=1), P(X|Y=0), P(Y|Z=1), P(Y|Z=0).
$$
These are given by the sensitivity and specificity values that you have measured or assumed. So more succinctly we know:
$$
P(X|Y), P(Y|Z)
$$
What we want to know is $P(X|Z)$, the true sensitivity and specificity of your classifier. We can obtain this from $P(X,Y|Z)$ by summing over all (both) possible values of $Y$, if we can get $P(X,Y|Z)$. It is a simple consequence of the definition of conditional probability that
$$
P(X,Y|Z) = P(X|Y,Z) \cdot P(Y|Z),
$$
[if this is new to you remove the Z and it will be quite familiar]. But we don't know $P(X|Y,Z)$. Therefore the pivotal assumption, without which I don't think we can do anything (unless you know the true label of individuals in which case you should train on that), is that $X,Z$ are conditionally independent, given $Y$, in which case $P(X|Y,Z) = P(X|Y)$, that is the only way the true label effects your prediction is by effecting the (gold) label you used to train your predictor.
So if your comfortable with that assumption, then we can proceed to calculate:
$$
P(X=1|Z=1) = P(X=1|Y=1)P(Y=1|Z=1) + P(X=1|Y=0)P(Y=0|Z=1) = 0.84\cdot0.95 + 0.05\cdot0.05
$$
I will leave the other calculation to you. Hope this was helpful!