Using N -body simulations we study the impact of various systematic effects on the low-order moments of the cosmic velocity field: the bulk flow (BF) and the cosmic Mach number (CMN). We consider two types of systematics: those related to survey properties and those induced by the observer's location in the Universe. In the former category we model sparse sampling, velocity errors, and survey incompleteness (radial and geometrical). In the latter, we consider local group (LG) analogue observers, placed in a specific location within the cosmic web, satisfying various observational criteria. We differentiate such LG observers from Copernican ones, who are at random locations. We report strong systematic effects on the measured BF and CMN induced by sparse sampling, velocity errors and radial incompleteness. For BF most of these effects exceed 10% for scales R ≲100 h-1 Mpc . For CMN some of these systematics can be catastrophically large (i.e., >50 %) also on bigger scales. Moreover, we find that the position of the observer in the cosmic web significantly affects the locally measured BF (CMN), with effects as large as ∼20 % (30 % ) at R ≲50 h-1 Mpc for a LG-like observer as compared to a random one. This effect is comparable to the sample variance at the same scales. Such location-dependent effects have not been considered previously in BF and CMN studies and here we report their magnitude and scale for the first time. To highlight the importance of these systematics, we additionally study a model of modified gravity with ∼15 % enhanced growth rate (compared to general relativity). We found that the systematic effects can mimic the modified gravity signal. The worst-case scenario is realized for a case of a LG-like observer, when the effects induced by local structures are degenerate with the enhanced growth rate fostered by modified gravity. Our results indicate that dedicated constrained simulations and realistic mock galaxy catalogs will be absolutely necessary to fully benefit from the statistical power of the forthcoming peculiar velocity data from surveys such as TAIPAN, WALLABY, COSMICFLOWS-4 and SKA.