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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 darkenergy and modified gravity: a cosmological constant, a perfect fluid withconstant equation of state $w$, quintessence (minimally coupled scalar field),MOND, and symmetrons (coupled scalar field). We explore generic coupled scalarfield theories, with the symmetron model as an explicit example. We find thattheories which attempt to eliminate dark matter by fitting rotation curvesproduce mass estimates in the timing argument which are not compatible with theluminous mass of the galaxies alone. Assuming that the galaxies are approachingtheir 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 beobtained in the case of a previous fly-by event between the MW and M31. Thesymmetron model suggests a mass too high to be explained without additionaldark matter ($\mathcal{O}(10^{12}) M_\odot$), suggesting that there is amissing mass problem in this model. We also demonstrate that tensions inmeasurements of $H_0$ can produce an uncertainty in the Local Group massestimate comparable to observational uncertainties on the separation andrelative 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.

The Two Body Problem in the Presence of Dark Energy and Modified Gravity: Application to the Local Group