The galaxy NGC 2770 hosted two core-collapse supernova (SN) explosions, SN 2008D and SN 2007uy, within 10 days of each other and 9 years after the first SN of the same type, SN 1999eh, was found in that galaxy. In particular, SN 2008D attracted a lot of attention due to the detection of an X-ray outburst, which has been hypothesized to be caused by either a (mildly) relativistic jet or the SN shock breakout. We present an extensive study of the radio emission from SN 2008D and SN 2007uy: flux measurements with the Westerbork Synthesis Radio Telescope and the Giant Metrewave Radio Telescope, covering similar to 600 days with observing frequencies ranging from 325 MHz to 8.4 GHz. The results of two epochs of global Very Long Baseline Interferometry observations are also discussed. We have examined the molecular gas in the host galaxy NGC 2770 with the Arizona Radio Observatory 12 m telescope, and present the implications of our observations for the star formation and seemingly high SN rate in this galaxy. Furthermore, we discuss the near-future observing possibilities of the two SNe and their host galaxy at low radio frequencies with the Low Frequency Array.