How do OGLE-III and GAIA measure the mass of free microlensing black holes?

The introduction of the Wyrzykowski & Mandel paper gives the following information about estimating the lens mass.

In order to obtain the mass of the lens (Gould 2000a), it is necessary to measure both the angular Einstein radius of the lens ($theta_mathrm{E}$) and the microlensing parallax ($pi_mathrm{E}$)

$$M = frac{theta_mathrm{E}}{kappa pi_mathrm{E}}$$

where $kappa = 4G / (c^2 mathrm{AU}) = 8.144 mathrm{mas/M_odot}$; and $pi_mathrm{E}$ is the length of the parallax vector $mathbf{pi_mathrm{E}}$, defined as $pi_mathrm{rel}/theta_mathrm{E}$, where $pi_mathrm{rel}$ is relative parallax of the lens and the source. The microlensing parallax vector $mathbf{pi_mathrm{E}}$ is measurable from the non-linear motion of the observer along the Earth’s orbital plane around the Sun. The effect of microlensing parallax often causes subtle deviations and asymmetries relative to the standard Paczynski light curve in microlensing events lasting a few months or more, so that the Earth’s orbital motion cannot be neglected. The parameter $mathbf{pi_mathrm{E}}$ can also be obtained from simultaneous observations of the event from the ground and from a space observatory located ∼1 AU away (e.g., Spitzer or Kepler, e.g., Udalski et al. 2015b, Calchi Novati et al. 2015, Zhu et al. 2017).

In particular, the Gould 2000a paper gives a good summary of the various relationships between the quantities. The Udalski et al. 2015b notes that the distance between the Earth and Spitzer (which would also apply to Gaia) means that Spitzer would see differences in the light curve, allowing the parallax to be determined.

Note that things get more complicated if the source is a binary, in which case a "reverse parallax" effect from the source's orbital motion, usually called "xallarap" needs to be taken into account — but that's a matter for another question...

The other relevant quantity is the angular Einstein radius of the lens. In their discussion of measuring $theta_mathrm{E}$, Wyrzykowski & Mandel reference Rybicki et al. 2018. That paper notes that precision astrometry can help measure $theta_mathrm{E}$ because microlensing also changes the apparent position of the source:

The positional change of the centroid depends on the $theta_mathrm{E}$ and separation $u$. Contrary to the photometric case, the maximum shift occurs at $u_0 = sqrt{2}$ and reads (Dominik & Sahu 2000)

$$delta_mathrm{max} = frac{sqrt{2}}{4} theta_mathrm{E} approx 0.354 theta_mathrm{E}$$

Thus, for the relatively nearby lens at $D_l = 4 mathrm{kpc}$, source in the bulge $D_s = 8 mathrm{kpc}$ and lensing by a stellar BH with the mass $M = 4M_odot$, the astrometric shift due to microlensing will be about 0.7 milliarcsecond.

The bulk of the paper goes on to determine that these shifts should be observable by Gaia.

Another way to measure the size of the lens is to measure the lens-source proper motion by searching for the lens several years after the event, this has been done for a couple of exoplanet-hosting lenses but would not be possible for a dark lens like a black hole.