  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%), IIb (12%), and IIn (9%) 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 days. We also discuss the LFs in different galaxy sizes and inclinations,
and for different SN subclasses. Some notable results are ... (abridged).
