They most certainly purr in the company of other cats. They purr in various circumstances too. It is hard to get information on the purring of solitary cats; I can't find a record of experiments to assess the probability that they do to any non-trivial extent. Purring is of course a social signal, but then so is human speech, and many humans speak to themselves in solitude, so possibly so do some cats purr similarly in solitude.  I should not be surprised to find that it happens occasionally.

Mind you, in direct contradiction to most people's impression, cats hate being left alone for any significant period; it is wanton cruelty to do it to any domestic pet. (Leaving it alone with a favoured companion member of the household, such as another cat or a friendly dog, does not count as solitude for reasonable periods.) 

Anyway, a cat in solitude is likely to be too unhappy to purr much; soliliquising or not.