Let me simply start by saying: in my opinion they should.
I know how linguists may react- linguistics is the study of language, the word "linguist" should not be confused with "polyglot", there are non-linguist polyglots and non-polyglot linguists. I know, too, that linguists deal with phonology, morphology, lexis, syntax, grammar, semantics, pragmatics, language history, etc.
But personally I don't trust linguists who don't know another language, not that they don't speak another language fluently but when they don't understand anything at all but their mother tongue, because of the view, supported by some personal experience, that you do not really know your language until you have known another one. Languages are very different, especially when they don't belong to the same group. Chances are, the rules in your mother tongue, you take for granted and see as understandable, unquestionable and more normal than anything in the world, until you learn another language that challenges everything you have held as truths, makes your belief collapse and tears your world apart. Of course, you may stand in a distance and hear somebody else speak about a language that is very different from yours, you may hear about certain rules in that language that sound incomprehensible to you, unusual, unthinkable, but you're too far away from it to accept it, understand it, be affected by it. Learning another language expands you perspective and gives you insight into what you've never really known before about your mother tongue. Speaking it makes you think in it.
And even though so far I've been talking about a simple "another language", it's a better idea to know a language that belongs to another group, the way I, a Vietnamese speaker, speak English and now learn Norwegian and in the future will learn another language that isn't Germanic. Truly knowing something and hearing it from others are not the same. Especially when a linguist makes claims and generalisations about languages in general though they themselves are monolingual, I can't take them seriously.
Of course, one may argue still that linguists aren't required to speak many languages and it makes no difference. Of course, one may say my point is invalid, especially because I'm not a linguist myself.
But that's my 2 cents on the subject.