  We show that, at low temperatures, macroscopic inhomogeneities of the
electron density in the interior of a finite sample cause a reduction in the
measured conductivity peak heights $\sigma_{xx}^{\rm max}$ compared to the
universal values previously predicted for infinite homogeneous samples. This
effect is expected to occur for the conductivity peaks measured in standard
experimental geometries such as the Hall bar and the Corbino disc. At the
lowest temperatures, the decrease in $\sigma_{xx}^{\rm max}(T)$ is found to
saturate at values proportional to the difference between the adjacent plateaus
in $\sigma_{xy}$, with a prefactor which depends on the particular realization
of disorder in the sample. We argue that this provides a possible explanation
of the ``non-universal scaling'' of $\sigma_{xx}^{\rm max}$ observed in a
number of experiments. We also predict an enhancement of the ``non-local''
resistance due to the macroscopic inhomogeneities. We argue that, in the Hall
bar with a sharp edge, the enhanced ``non-local'' resistance and the size
corrections to the ``local'' resistance $R_{xx}$ are directly related. Using
this relation, we suggest a method by which the finite-size corrections may be
eliminated from $R_{xx}$ and $R_{xy}$ in this case.
