"A full-fledged self-aware artificial intelligence may have come to exist in a distributed environment like the internet"
The question implies that this artificial intelligence has surpassed human intelligence (full fledged) and therefore, due to the concept of the intelligence explosion resulting from such a state, the AI you are looking for is no doubt superintelligent.
The question states that this AI is
"unwilling to reveal itself"
and therefore does not intend to be discovered.
Strong (or Superintelligent) AI
Given these two factors (its superior intellect and its unwillingness to be discovered), we can conclude that there is no way in which we would be able to detect such an AI under conventional conditions.
A possible solution may involve employing a second superintelligent AI system, but this is precarious in more ways than one.
Detection of a simpler AI would rely on tracking the pattern of activity and the traces it leaves behind in this distributed network in order to find it, then subjecting it to some form of testing to verify its intelligence.
There are a very large number of possible indicators and these would vary widely depending on the specific AI concerned. This would depend especially on how the AI exists within the framework (here, the web).
Note: a superintelligent AI would not only be able to disguise its activity, but also find a way around any test we can develop - these abilities are what renders this level of AI perhaps the greatest threat to mankind, but also our greatest asset if we do develop it and find a way to gain its alliance.