The whole point of protests is to allow the people to express problems by peaceful gatherings to bring up an issue into evidence. However, such gatherings of people can also prove a serious threat for the security of the people involved themselves. The idea is that you have police monitor the situation, and in case something goes out of hand and a protest turns into a riot is quenched.

Where the serious issue is on who, and why uses a protest as a political tool. For example; in Greece it is well-known that certain people involved in causing the riots have close ties with certain parties to induce violence from an otherwise peaceful protest. Perhaps, since inciting a riot from a protest isn't a terribly hard thing to do. (Especially when the issue is a grave one)

On whether actual protesting should be allowed to the general populace is because the basis of a democracy consists on expression your own opinions, and concerns. Unlike an autocracy, or other systems that do view such an expression as a threat (since the decision-making and awareness rests on the ruler(s)) a driving factor behind a democracy stems from the ability of people being able to express themselves.

That's also why it's also frowned upon when a peaceful protest is reprimanded by various police forces. Who later may proceed to do some damage control at various lengths with different degrees of success.