Luminous compact blue galaxies (LCBGs) were a factor of ten more common at z ~ 1 than they are in the local universe. This discrepancy means that these galaxies must evolve quickly. Even though we know that they must have high star formation rates over a short timescale before their star formation rapidly declines, it is not known why these galaxies stop forming stars, nor is it know what their descendants look like today. We have combined new and archival radio continuum and far-infrared observations of local LCBGs to characterize their global star formation properties. The LCBGs in our sample all have evidence of ongoing star formation, but appear to be heterogeneous in their star formation properties and ages. We find that LCBGs with clumpy appearances appear to have a range of star formation rates and ages, while LCBGs that do not have prominent star-forming clumps tend to have shorter star formation timescales on average. Thus, it is likely that the physical causes of star formation in LCBGs with and without clumpy appearances are diverse.