Aims: We derive the probability density functions (PDFs) of column density for a complete sample of prominent molecular cloud complexes closer than 200 pc. Methods: We derive near-infrared dust extinction maps for 23 molecular cloud complexes, using the "nicest" colour excess mapping technique and data from the 2MASS archive. The extinction maps are then used to examine the column density PDFs in the clouds. Results: The column density PDFs of most molecular clouds are well-fitted by log-normal functions at low column densities (0.5 mag < A_v < 3-5 mag). However, at higher column densities prominent, power-law-like wings are common. In particular, we identify a trend among the PDFs: active star-forming clouds always have prominent non-log-normal wings. In contrast, clouds without active star formation resemble log-normals over the whole observed column density range, or show only low excess of higher column densities. This trend is also reflected in the cumulative PDFs, showing that the fraction of high column density material is significantly larger in star-forming clouds. These observations are in agreement with an evolutionary trend where turbulent motions are the main cloud-shaping mechanism for quiescent clouds, but the density enhancements induced by them quickly become dominated by gravity (and other mechanisms) which is strongly reflected by the shape of the column density PDFs. The dominant role of the turbulence is restricted to the very early stages of molecular cloud evolution, comparable to the onset of active star formation in the clouds.