I would not say absolutely no, but since they are an indicator we used in water quality testing for sanitation, it would not be a very good indicator if they are multiplying in polluted water.  If you find higher concentrations as you go downstream with more flow, you have to address are their other sources, or is the sampling system biasing the data.  I have one brief report I coauthored concerning coliform from Stekoa Creek into the Chatttooga River about in 1994 I think.  In watching sediment entry also from Stekoa Creek, I learned that it may take several meander bends and turbulence for water entering from a tributary is mixed downstream.  If you sample from a stream's edge and near an area of fecal entry or concentration, you may find unusually high numbers locally within the stream or river that are not mixed completely yet in the discharge of that stream or river.  Streams or rivers with relative uniform flow pattern and velocity such as extended runs and wide and shallow such as Rosgen F type streams as the Chattooga River may take extended distances to mix.  For some sampling schemes, even though we prefer or it is convenient to sample from the margins or edge, obtaining sampling devices and taking a cross section with varying depths is probably the main way of insuring you get a quality sample.  Sometimes this might mean sampling from a boat and lowering the sampler into the water to its bottom and back up.  There are various hand and cable devices to accomplish this level of detail.  If you have a fluorometer, you might try some dye testing with fluorencent dye from point of entry to various downstream areas to find out where mixing occurs.  Otherwise, visually if there is sediment and a point of entry, go at least downstream far enough in sampling so you cannot visually see difference in sediment loading across the water, and then maybe go further if logical to do before the next tributary or pollution source.  Because animals can contribute fecal material, that is a concern.  I am attaching a file on drinking water in forests and grasslands, but it has some great info on fecal coliform and other contaminants.  It may give you some ideas.  In the back of my mind, I remember someone saying the idea that multiplication of fecal organism is a possibility, but don't remember reading about it or specifically that it is an issue.

Like the answer above, we cannot be absolutely sure that they do not reproduce in the water column.  There are literature that suggest the possibility of fecal coliforms persisting in biofilms and others suggesting the possibly of replication in the sediment of lakes and rivers. When the sediments are disturbed, the fecal coliforms are then detectable in the water column.

Here is one article that you may find helpful.

This is an old article that might help shed some light on the question.  http://www.ncbi.nlm.nih.gov/pmc/articles/PMC380575/

This article may be even more helpful.

http://aem.asm.org/content/76/19/6477.full

Yes, it can grow in river water, particularly like river water of Ganges & similar other rivers, which is highly polluted in downstream & often contains floating/submerged animal carcasses and similar that might facilitate growth.