I remember asking my DM this question, but I'd like to build on it using RAW terms.
Intelligence Checks are:
An Intelligence check comes into play when you need to draw on logic,
education, memory, or deductive reasoning. The Arcana, History,
Investigation, Nature, and Religion skills reflect aptitude in certain
kinds of Intelligence checks.
This does give the information that, although intelligence checks are useful when trying to piece an observation together, they are based on what information you can gather in that scene and how much of it you can relate to the knowledge you already know. Though they can give you as much deductive reasoning as Sherlock Holmes, it is actually opinion based and far from infallible.
Likewise, though in the PHB, it says:
Beginning at 1st level, you have significant experience studying,
tracking, hunting, and even talking to a certain type of enemy.
the actual experiences you have is specific to your character, such as did they spend their experience studying their enemy through observing their war camps or their villages? or even did you spend most of your time communicating with them through war banters, in drunken tavern talks, or spent time negotiating peace with them through months of trying to understand their side of the story? In this way, even childhood friends who turned ranger with the same favored enemy can have different experiences and understandings about their enemies.
Another major aspect to consider is that, though you have great deduction skill, this is still based on observation. That said, Sherlock made many investigation checks to draw his conclusion, and likewise, your intelligence check can only be accurate up to the investigations you did.
To elaborate, say you approached a couple of giants talking around a fire. If you only use your ability to understand their language, you might think their banter about how best to cook a human might be talking about a future war, though if you investigate the fire, you might notice that there's a family tied to a log being rotated over said fire. Further investigation on the surrounding might show that this family is actually a pack of thieves that fell under the misfortune of crossing giants while fleeing from their last job. Even further investigation might be that the human in question is actually wanted for murder dead or alive with a large bounty on his head and the items he stole were from your home town's local nobles.... but without doing each investigation checks individually, your skills at deduction are limited to overhearing a conversation.
Putting it all together.
After talking to my DM who expressed me similar to what I just said, and further investigating into deductive reasoning to find out that even Sherlock Holmes was only ever accurate for certain crimes, and would fail as a private detective observing if your wife is cheating on you (as an example), I think it's unfair to say that it's up to the DM to determine what information they should give the character you're giving life to. Rather, I would say that the best solution is to work with your DM by offering your thoughts on why and what your trying to notice and recall. This also avoids long-winded recollections such as Giants love to sit and gossip by open fires. They tend to enjoy that more than going to human pubs. They also enjoy teasing each other about their heights, and............ since that is also things your player can recall observing.
As such, as a player, I found it best to answer a few questions myself before calling for an intelligence check so that I assure that my quest is on the same thought path as my DM.
- What is my end goal? What am I trying to deduce from the situation?
- What part of my enemy's lifestyle have I observed in the past? What part of it is relevant to this situation?
- What observations can I make to best be equipped to make a logical assumption that results in answering my first question?
With those answered, I express to my DM that I'd like to make an intelligence check on my favored enemies based on x,y, and z investigations to see if a,b, or c, is happening as I have experience dealing with these beings in 1,2,and 3, situations.
Example based on before and my actual experience. I would like to make an intelligence check on the giants who are my favored enemies based on observing their campsite and conversation to gain enough information to logically deduce why they are here, if they are on guard or easily avoidable, and if I can gather useful info about future plans that I can use to warn others with. Since I've had experiences observing their war and hunting parties in my training, I feel I'm well equipped to gather that much information.
Also remember that you can check your observations with your DM if you feel you missed some clue in your investigation, such as "I'd like to gather a better understanding by trying to peak at their loot".
Hope that helps.