A detective was a person who investigated crimes so as to solve them, usually as part of an official police force. A private investigator was a freelance detective.
Duggan was a detective hired to investigate a plot to steal the Mona Lisa from the Louvre in Paris in 1979. (TV: City of Death)
Avan Tarklu, also known as Frobisher, was a Whifferdill private investigator who first met the Sixth Doctor while investigating him. (COMIC: The Shape Shifter)
Inspector Lestrade was a Scotland Yard detective. (PROSE: All-Consuming Fire)
In the Victorian era, in London, the Silurian Vastra was known as "the Great Detective" (HOMEVID: The Great Detective) for solving crimes, sometimes alongside the Doctor, one of her oldest friends. (TV: The Snowmen, The Crimson Horror, Deep Breath)
The Sleeze Brothers were human private investigors who met the Monk and the Seventh Doctor. (COMIC: Follow That TARDIS!)
In 1920, detective Merrivale Carr investigated the apparent murders of Sebastian Marten and Paul Steiffers at Lord Farthingale's country retreat. The Ninth Doctor and Rose helped in the investigation. (PROSE: Doctor vs Doctor)
The murders of several amateur detectives were investigated by a man and Georgie Price Jones with the aid of Iris Wildthyme and Tom. (PROSE: The Sleuth Slayers)