Recent discovery of many dwarf satellite galaxies in the direction of the Small and Large Magellanic Clouds (SMC and LMC) provokes questions of their origins, and what they can reveal about galaxy evolution theory. Here, we predict the satellite stellar mass function of Magellanic Cloud-mass host galaxies using abundance matching and reionization models applied to the Caterpillar simulations. Specifically focusing on the volume within 50 kpc of the LMC, we predict a mean of four to eight satellites with stellar mass $M_{\rm star} > 10^4 M_\odot$, and three to four satellites with $80 < M_{\rm star} \leq 3000 M_\odot$. Surprisingly, all 12 currently known satellite candidates have stellar masses of $80 < M_{\rm star} \leq 3000 M_\odot$. Reconciling the dearth of large satellites and profusion of small satellites is challenging and may require a combination of a major modification of the $M_{\rm star}-M_{\rm halo}$ relationship (steep, but with an abrupt flattening at $10^3 M_\odot$), late reionization for the Local Group ($z_{\rm reion} \leq 9$ preferred) and/or strong tidal stripping. We can more robustly predict that $\sim53$ per cent of satellites within this volume were accreted together with the LMC and SMC and $\sim47$ per cent were only ever Milky Way satellites. Observing satellites of isolated LMC-sized field galaxies is essential to place the LMC in context, and to better constrain the $M_{\rm star}-M_{\rm halo}$ relationship. Modelling known LMC-sized galaxies within 8 Mpc, we predict 1-6 (2-12) satellites with $M_{\rm star} > 10^5 M_\odot (M_{\rm star} > 10^4 M_\odot)$ within the virial volume of each, and 1-3 (1-7) within a single 1.5° diameter field of view, making their discovery likely.