  We explore mass estimation of the Local Group via the use of the simple,
dynamical `timing argument' in the context of a variety of theories of dark
energy and modified gravity: a cosmological constant, a perfect fluid with
constant equation of state $w$, quintessence (minimally coupled scalar field),
MOND, and symmetrons (coupled scalar field). We explore generic coupled scalar
field theories, with the symmetron model as an explicit example. We find that
theories which attempt to eliminate dark matter by fitting rotation curves
produce mass estimates in the timing argument which are not compatible with the
luminous mass of the galaxies alone. Assuming that the galaxies are approaching
their first encounter, MOND gives of around $2.7\times 10^{10} M_\odot$,
roughly 10\% of the luminous mass of the LG, although a higher mass can be
obtained in the case of a previous fly-by event between the MW and M31. The
symmetron model suggests a mass too high to be explained without additional
dark matter ($\mathcal{O}(10^{12}) M_\odot$), suggesting that there is a
missing mass problem in this model. We also demonstrate that tensions in
measurements of $H_0$ can produce an uncertainty in the Local Group mass
estimate comparable to observational uncertainties on the separation and
relative velocity of the galaxies, with values for the mass ranging from $4.5 -
5.4 \times 10^{12} M_{\odot}$ varying $h$ between 0.67 and 0.76.
