This is the second paper of a series in which we present new measurements of the observed rates of supernovae (SNe) in the local Universe, determined from the Lick Observatory Supernova Search (LOSS). In this paper, a complete SN sample is constructed, and the observed (uncorrected for host-galaxy extinction) luminosity functions (LFs) of SNe are derived. These LFs solve two issues that have plagued previous rate calculations for nearby SNe: the luminosity distribution of SNe and the host-galaxy extinction. We select a volume-limited sample of 175 SNe, collect photometry for every object and fit a family of light curves to constrain the peak magnitudes and light-curve shapes. The volume-limited LFs show that they are not well represented by a Gaussian distribution. There are notable differences in the LFs for galaxies of different Hubble types (especially for SNe Ia). We derive the observed fractions for the different subclasses in a complete SN sample, and find significant fractions of SNe II-L (10 per cent), IIb (12 per cent) and IIn (9 per cent) in the SN II sample. Furthermore, we derive the LFs and the observed fractions of different SN subclasses in a magnitude-limited survey with different observation intervals, and find that the LFs are enhanced at the high-luminosity end and appear more ‘standard’ with smaller scatter, and that the LFs and fractions of SNe do not change significantly when the observation interval is shorter than 10 d. We also discuss the LFs in different galaxy sizes and inclinations, and for different SN subclasses. Some notable results are that there is not a strong correlation between the SN LFs and the host-galaxy size, but there might be a preference for SNe IIn to occur in small, late-type spiral galaxies. The LFs in different inclination bins do not provide strong evidence for extreme extinction in highly inclined galaxies, though the sample is still small. The LFs of different SN subclasses show significant differences. We also find that SNe Ibc and IIb come from more luminous galaxies than SNe II-P, while SNe IIn come from less luminous galaxies, suggesting a possible metallicity effect. The limitations and applications of our LFs are also discussed.