Saying you should produce one or several outputs randomly without further specification
One random output
Say someone writes a challenge about generating a labyrinth of a given size (width and height) taken as input, and they say that you should create a randomly arranged labyrinth of that size.
What does random mean here? As it has been specified, you may very well create two possible labyrinths that comform to the size specs and pick randomly between the two. Or even create just one and claim it is random.
As it stands, the challenge is not well specified. Just saying random means nothing. The probability distribution needs to be indicated. Some correct specifications would be (these are excluding alternatives):
All possible labyrinths of the given size should be produced with the same probability;
All possible labyrinths of the given size should have a nonzero probability of occurrence;
The code should be able to produce at least 10 different labyrinths of the given size, each with nonzero probability.
Option 1 corresponds to a uniform distribution, which is "as random as it gets" (but may be too demanding for certain challenges). Option 2 is more flexible on the answerer, and is probably a good default. With option 3, the challenge writer should be careful to check that it is always possible to generate at least 10 valid outputs.
Several random outputs
The problem is aggravated when there are several random pieces (numbers, characters, etc) that need to be produced. Consider for example a challenge that takes a string as input and asks to introduce a random printable ASCII character after each character of the input string.
First off, the same considerations as above apply (Can I choose from just "a", "b", "c" as my random characters? Do I need to pick among those characters with the same probability?). But in this case there's the additional problem of statistical relationship between the random outputs. Can I randomly pick a character and insert that same character in all required positions? Or do the inserted characters need to be independently picked?
To avoid this ambiguity, the possible relationship between the randomly generated pieces needs to be specified in the challenge. In more proper terms, the joint probability distribution of the random pieces needs to be specified. Sorry if it sounds a little complicated, but this is necessary.
If you are not sure and want to keep things simple, after having specified the distribution of each piece (as discussed above) you can say that the random pieces should be statistically independent. This means that each piece should be randomly generated without regard of the random choices that were made/will be made for the other pieces.
The point is to properly specify "randomness". Don't rely on "common sense" or assume that answerers will adhere to the "spirit" of the challenge. Your that's against the spirit of the challenge may very well be someone else's clever trick that saved me a few bytes.