In most cases, you would use a definite article before 'private sector'. This is because you are referring to a specific known sector i.e. the private sector as opposed to the public sector. An exception would be when the term 'private sector' is used adjectivally, as in "Private sector employees are paid more." 'Criminal activities' refers to criminal activities in general, as in "There has been a rise in criminal activities over recent years." 'The criminal activities' refers to the specific criminal activities associated with a particular known individual, group or location. The listener or reader knows what this context is because it has been mentioned or is understood, as in "The police had been monitoring the criminal activities of this gang for some time." By the way, this is basic grammar rather than a rule that's only applicable to certain compound nouns. If it's understood that you're referring to something specific, you use a definite article. "Do you like coffee?" means coffee in general, whereas "Do you like the coffee?" would mean that you're asking about the particular coffee that you're drinking at the moment, or the particular type of coffee served in the bar which you're currently discussing. I hope this helps.