My educated guess (=couln't find the reference book from my messy 'archives'

is that the basically military handled the military intelligence, the interior ministry the counter espionage and military and the foreign ministry took care of the espionage and related matters abroad.
On the military side, most notable names are colonels Reino Hallamaa and Aladar Paasonen. The former was the head of signals intelligence and the latter the chief of military intelligence. I believe both of them published memoirs, but I do not know if they are publicly available in any other language than finnish.
I do not remember reading anything detailed about the training operations - especially for espionage operations. There is lots of material available for long range patrols and signals intelligence, though.
Long range patrols were directlyy under the General HQ. Their missions were usually intelligence gathering and sometimes took them several hundreds of kilometres behind the enemy lines and lasted for several weeks. Divisional level patrols operated closer and were quite often used for destroying headquarters and supply dumps.
Internal security and counter espionage was mainly handled by the secret police (Valtiollinen poliisi = State Police). There is not much material available for their organisation, training etc. What is available is the successes they chose to publish (secret recording of Hitler's comments about the war against the Soviets in 1942, copying Himmler's documents in Helsinki, captures of Soviet trained spies and using a captured spy to report Helsinki as 'destroyed' after February 1944 bombing raids, just to mention some).
Radio and signals intelligence had a history from early 1920's. They had a lot of experience when the wars started 1939 and 1941. They employed a lot of mathematicians to perfect encryption techniques - in fact a lot of the Finnish documents and expertise was transferred to USA after the war.
Signals intelligence concentrated mainly on analysing and decrypting Soviet radio traffic. They were able to decrypt most tactical encryption codes and sometimes even the more secure higher level encrypted messages. They also worked on everything from listening into front line Soviet phone conversations to decrypting diplomatic messages.
During summer 1944 radio intelligence managed to provide accurate information for the defending Finnish forces - accurate enough that in some cases Soviet attacks were pre-empted by artillery barrages to the enemy headquarters and staging areas.
Almost the same organisations exist now. (From recent newspaper articles) The 'secret police' is Suojelupoliisi ('Protection police'

. Signals and radio intelligence are names Viestikoelaitos ('Communications experimental facility'

and Radiotutkimuslaitos ('Radio research facility'