The discussion about DHCP and the number of DHCP servers on the network is, on this occasion, in my opinion a red herring.
The "original" VM modems supported a single PC. This PC picked up an IP address in the public range from VM's servers.
When a router is placed between the VM modem and the PC, the PC translates the IP address from the (single) one which the modem is handing out to the router's default gateway, which can be 192.168.1.1 or any other private IP address. Your PC/printer/laptop/what-have-you can then be assigned (either through DHCP or manually) an IP address in the same range as the default gateway. All those machines will have their default gateway set to the IP address of the router, and as all those addresses are on the same subnet, the machines will be able to "see" each other and also "talk" to the internert via the router. And that's about it!