In order to determine the answer to this question you must first understand the concepts of the Shakespearean Clown and Fool. Both misuse language and comment on important issues of the day or characters of the play, while making a farce of the issues and characters. However, the Clown does this accidentally being an un- or under-educated person from the country who confuses words and twists ideas, while the Fool does this intentionally being a sophisticated, educated city dweller.
Dogberry is a Clown. His function is to illuminate important social issues or to provide insight into important characters. He does this by making a farce of the issue or character. An example of this is when he asks in III.iii who is "desertless" to be constable. This of course means who is unworthy to be constable which makes a farce of the policing system in English country seats.
Now, back to your question: Dogberry is at one and the same time silly and conceited as well as well-intentioned and honest. Let me sort out the seeming paradox. In his own person, the character of Dogberry is well-intentioned and honest and fully means to fulfill his duties with all faithfulness and earnestness. His accomplishments--in his view--in life have given him a sense of pride that reveals itself as a conceit in success.
Nonetheless, the fact that he is cast as a Clown means that Shakespeare manipulates this pride of accomplishment, this conceit, in this character in order to reveal, in this example, the failures of the public duties of country policing agencies. In order to do this, Shakespeare gives Dogberry the (unenviable) quality of knowing a dangerous amount of logic and the English language. These prove dangerous because Dogberry twists and mistakes them in such ways that any right thinking, well-educated person will find silly and laughable, though persons with less knowledge and experience than Dogberry will find his utterances impressive and masterful.
It is in these regards that Dogberry is silly and conceited while being honest and well-intentioned. The fact that he is perceived as both, and the fact that some viewers/readers of Shakespeare feel the conflict between the two descriptions shows how thoroughly and how well Shakespeare crafted Dogberry's intricate character. In the example below, the issue Shakespeare makes farce of is policing while the language and logic Dogberry twists are highlighted:
... You shall also make no noise in
the streets; for, for the watch to babble and to
talk is most tolerable and not to be endured.
We will rather sleep than talk: we know what
belongs to a watch.
Why, you speak like an ancient and most quiet
watchman; for I cannot see how sleeping should