Short Version: If a druid knows what animals are local to an area, and changes into something that wouldn't cause immediate suspicion (e.g., changing into a polar bear in the middle of the desert) without close inspection, would your average joe be able to tell it's a druid, or just think it's an animal?
A bit more detail: Say the druid transformed into a dog and was sitting by the side of a road just outside of town, watching for someone or something. Would anyone passing by instinctively know it's a druid? If not, what if they come up to simply pat it? At what point does someone go, "Hang on, something's not quite right about this dog"?
Similarly, the party is setting up an ambush and the druid is in plain sight in a form that wouldn't suggest immediate danger, again we'll go with a dog on the side of a road, would the ambushees have cause to draw swords at the sight of the dog or, in general, just shrug it off?
So far, I've handled this with a bluff check from the druid but it seems to me, without cause for suspicion, they should just seem like an ordinary animal.
Best Answer
Tl;dr - The Druid makes a Charisma (deception) check, an observer has to beat their result with a Wisdom (insight) check. Proficiency in Nature would allow a character who was not proficient to add their proficiency bonus to the whichever of the 2 checks they were making.
So, there are 2 questions here. The first is whether the Wild Shaped Druid is physically indistinguishable from a normal animal. Wild Shape says that you assume the form of a regular animal, not a Dire, celestial, fey, or anything else. So I think we can take it as read that if you Wild Shape into a dog, you look like a dog, because, well, on a physical level, you are a dog.
The second, more interesting question, is the behavioural issue. Can an intelligent humanoid act exactly like a regular animal? Given that you retain your mental scores and the shape you assume doesn't influence your actions in any way, it seems clear that physically being a dog doesn't make you automatically act like one. So our Druid has to pretend to be a dog. This works exactly like Alter Self, which lets you make yourself appear as a member of another race. If you used Alter Self to pretend to be the local ruler, it would be assumed by all that Charisma (deception) checks would be required to be convincing. Note that Alter Self doesn't give any benefit to this check - it just fixes your appearance for you.
Now, acting is covered by Charisma (Deception) checks. So the obvious call would be to have the Druid make a Charisma (deception) check, which any observer would have to make a Wisdom (Insight) check to see through. This would probably work, but there is also Intelligence (Nature) to consider. Does the observer know how an animal of that type normally behaves?
There are a few options here. The first is to say that the observer has to make both an Intelligence (nature) check and a Wisdom (insight) check. The second is that the observer either has to make a Wisdom (insight) check or an Intelligence (nature) check. I would say that knowing how an animal normally behaves isn't the same as noticing that it is behaving oddly. Therefore, if I was going to include an Intelligence (nature) check, I would require it in addition to a Wisdom (insight) check.
On the other hand, I don't know much about dogs, but I could probably notice if one was, e.g., carefully watching everyone who went past. So I would say that Wisdom (insight) is all that is required. Perhaps an Intelligence (nature) check could be used to gain advantage on the Wisdom (insight) check, or proficiency in Nature could allow a character not proficient in Insight to add their proficiency bonus to the Wisdom roll. That last is what I'd rule.
Whatever decision you use for Nature to influence the Wisdom check, for the sake of both fairness and consistency the Druid should have the same benefit to their Charisma check. So I would allow a Druid proficient in Nature to add their proficiency bonus to the Charisma check even if they weren't proficient in Deception.