Suppose that someone (non-EU citizen) has permanent residency in one EU country. Can she live in another EU country? What are the risks? If the police say something, can she show the permanent residency card? Would there no basically be no problems as long as that person does not want to work or use the public service of the second country?
Directive 2003/109/EC affords "long-term residents" a right to reside in other EU countries under certain conditions, although three EU countries have opted out of this directive (the UK, Ireland, and Denmark). The answer to your question therefore depends on the EU countries involved. For the remainder of this answer, I assume that both countries have implemented the directive.
What are the risks?
There are no legal risks as long as the person complies with the requirements of the directive. If you're asking about a resident moving to another country without complying with the directive, the risks could be more significant, including the possibility of being deported back to the country of citizenship.
If police say something she can show the permanent residency card. So basically should not be a problem if that person does not want to work or use the public service in the second country right?
Showing the police in one EU country a permanent residency card for another EU country would satisfy them if they think the person has been traveling for 90 days or less. But if they have reason to suspect that the person has been there for longer, the card won't be of much help.