  Understanding planet formation is important in the context of the origin of
planetary systems in general and of the Solar System in particular, as well as
to predict the likelihood of finding Jupiter, Neptune, and Earth analogues
around other stars. We aim to precisely determine the radii and dynamical
masses of transiting planets orbiting the young M star AU Mic using public
photometric and spectroscopic datasets. We characterise the stellar activity
and physical properties (radius, mass, density) of the transiting planets in
the young AU Mic system through joint transit and radial velocity fits with
Gaussian processes. We determine a radius of $R^{b}$= 4.79 +/- 0.29 R$_\oplus$,
a mass of $M^{b}$= 9.0 +/- 2.7 M$_\oplus$, and a bulk density of $\rho^{b}$ =
0.49 +/- 0.16 g cm$^{-3}$ for the innermost transiting planet AU Mic b. For the
second known transiting planet, AU Mic c, we infer a radius of $R^{c}$= 2.79
+/- 0.18 R$_\oplus$, a mass of $M^{c}$= 14.5 +/- 3.4 M$_\oplus$, and a bulk
density of $\rho^{c}$ = 3.90 +/- 1.17 g cm$^{-3}$. According to theoretical
models, AU Mic b may harbour an H2 envelope larger than 5\% by mass, with a
fraction of rock and a fraction of water. AU Mic c could be made of rock and/or
water and may have an H2 atmosphere comprising at most 5\% of its mass. AU Mic
b has retained most of its atmosphere but might lose it over tens of millions
of years due to the strong stellar radiation, while AU Mic c likely suffers
much less photo-evaporation because it lies at a larger separation from its
host. Using all the datasets in hand, we determine a 3$\sigma$ upper mass limit
of $M^{[d]}\sin{i}$ = 8.6 M$_{\oplus}$ for the AU Mic 'd' TTV-candidate. In
addition, we do not confirm the recently proposed existence of the planet
candidate AU Mic 'e' with an orbital period of 33.4 days.
