I was expecting that you might have some. Do you?
Read this quick as most of it is off topic and will no doubt be redacted.
About the same as you I would imagine. There is plenty of information on the net but it is a tedious question of separating sheep from goats and most of us have better things to do; indeed just other things to do.
With regard to whether or not running a private investigation parallel with a police criminal investigation is illegal in Portugal who knows? I would imagine if one wrote to a Portuguese Advogado one would obtain a definitive answer at a price.
The nearest we have is the suggestion in Kate McCann's book that were doing something "technically illegal" and the article in the Daily Telegraph that I have already posted a link to.
According to Wikipedia which is not renowned for its accuracy:
"In some countries, the practice of a detective is not yet recognized in courts and judicial processes. One of these countries is Portugal, where the proof presented loses all the significance when collected by a private detective. Even under this circumstance, the practice of this activity is in demand, and ruled by a code of conduct".
Whether or not the practice is illegal, private investigators everywhere can run foul of obstructing the course of justice. In the United States private investigators have been subject to rigorous examination in knowledge of the law and investigation before being granted a licence. In 2007 in the United Kingdom every man jack could set themselves up as PI without requiring a licence as there was no licensing arrangement then; that may still apply but in 2013 the Home Sec. was proposing a system in the UK similar to the US.