We consider a spectral sharing problem in which a statistical (or widely distributed) multiple-input multiple-output (MIMO) radar and an in-band full-duplex (IBFD) multi-user MIMO (MU-MIMO) communications system concurrently operate within the same frequency band. Prior works on joint MIMO-radar-MIMO-communications (MRMC) systems largely focus on either colocated MIMO radars, half-duplex MIMO communications, single-user scenarios, omit practical constraints (clutter, uplink [UL]/downlink [DL] transmit powers, UL/DL quality-of-service, and peak-to-average-power ratio), or MRMC co-existence that employs separate transmit/receive units. The purpose of this and companion papers (Part II and III) is to co-design an MRMC framework that addresses all of these issues. In this paper, we propose signal processing for a distributed IBFD MRMC, where radar receiver is designed to additionally exploit the downlink communications signals reflected from a radar target. Extensive numerical experiments show that our methods improve radar target detection over conventional codes and yield a higher achievable data rate than standard precoders. The following companion paper (Part II) describes the theory and procedure of our algorithm to solve the non-convex design problem. The final companion paper (Part II) considers the case of multiple targets and examines the tracking performance of our MRMC system.