In this paper, we present the results of timing observations of PSRs J1949+3106 and J1950+2414, two binary millisecond pulsars discovered in data from the Arecibo ALFA pulsar survey (PALFA). The timing parameters include precise measurements of the proper motions of both pulsars, which show that PSR J1949+3106 has a transversal motion very similar to that of an object in the local standard of rest. The timing also includes measurements of the Shapiro delay and the rate of advance of periastron for both systems. Assuming general relativity, these allow estimates of the masses of the components of the two systems; for PSR J1949+3106, the pulsar mass is $M_p , = , 1.34^{+0.17}_{-0.15} , M_{odot}$ and the companion mass $M_c , = , 0.81^{+0.06}_{-0.05}, M_{odot}$; for PSR J1950+2414 $M_p , = , 1.496 , pm , 0.023, M_{odot}$ and $M_c , = , 0.280^{+0.005}_{-0.004}, M_{odot}$ (all values 68.3 % confidence limits). We use these masses and proper motions to investigate the evolutionary history of both systems: PSR J1949+3106 is likely the product of a low-kick supernova; PSR J1950+2414 is a member of a new class of eccentric millisecond pulsar binaries with an unknown formation mechanism. We discuss the proposed hypotheses for the formations of these systems in light of our new mass measurements.
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