We calculate analytically the probability distribution for peculiar velocities on scales from 10h/sup -1/ to 60h/sup -1/ Mpc with cosmic string loops as the dominant source of primordial gravitational perturbations. We consider a range of parameters .beta.G.mu. appropriate for both hot (HDM) and cold (CDM) dark matter scenarios. An .cap omega. = 1 CDM Universe is assumed with the loops randomly placed on a smooth background. It is shown how the effects can be estimated of loops breaking up and being born with a spectrum of sizes. It is found that to obtain large scale streaming velocities of at least 400 km/s it is necessary that either a large value for .beta.G.mu. or the effect of loop fissioning and production details be considerable. Specifically, for optimal CDM string parameters G.mu. = 10/sup -6/, .beta. = 9, h = .5, and scales of 60h/sup -1/ Mpc, the parent size spectrum must be 36 times larger than the evolved daughter spectrum to achieve peculiar velocities of at least 400 km/s with a probability of 63%. With this scenario the microwave background dipole will be less than 800 km/s with only a 10% probability. The string induced velocity spectrum is relatively flat out to scales of about 2t/sub eq//a/sub eq/ and then drops off rather quickly. The flatness is a signature of string models of galaxy formation. With HDM a larger value of .beta.G.mu. is necessary for galaxy formation since accretion on small scales starts later. Hence, with HDM, the peculiar velocity spectrum will be larger on large scales and the flat region will extend to larger scales. If large scale peculiar velocities greater than 400 km/s are real then it is concluded that strings plus CDM have difficulties. The advantages of strings plus HDM in this regard will be explored in greater detail in a later paper. 27 refs., 4 figs., 1 tab.