When you start learning a new language and reading all the motivation and instructional blogs, you find that there are a lot of strong opinions on whether or not it is important to speak correctly. Today, hopefully, I can help you cut through the dogma and the loud voices to figure out the answer for yourself.
In the early stages of learning a language, the most important activity is speaking: forcing recall from your brain, forming sounds and accents, creating thoughts. The biggest language learning progress is made during real use of the language.
So it's no surprise that a lot of people feel that it's not important to speak properly. You'll usually see this worded in more extremist ways, because people think it makes the case better: "it's okay to make a few mistakes", "you don't have to be perfect", "even natives make mistakes from time to time", etc.
When you're defending such an extremely isolated position, it's really hard to be wrong. But it's really unnecessary. The real benefit isn't to be "as error-prone as a native", right? The benefit is that allowing yourself to make mistakes gets you talking, which gets you learning, and that's a pretty good benefit.
Fluency and beyond
Later on in your language-learning progress, however, when you've established your fluency, people stop seeing you as someone who's trying really hard to learn the language, and they start accepting you as someone who speaks the language.
At that point, there is a mental shift. Instead of giving the benefit of the doubt, and patiently working with you to understand what you're saying, instead, they start just having conversations with you... you know, as an equal. But that's when your mistakes start to really show.
In spite of what people tell you, natives really don't make very many mistakes. They have occasional moments when their brain is going faster than their tongue (just like you have in your native language). They have expressions which are explicitly used in spite of being incorrect, such as slang expressions. But mistakes are actually very rare.
And what's more, when you make a simple mistake as a foreigner, you're usually saying something wrong that an average five-year-old always gets right. The effect is that whether you like it or not, whether it's conscious or not, whether people intend to or not, they're judging you by your speech.
Unlearn what is learned?
Here's where the dilemma comes. All those mistakes you allow yourself to make in the beginning form habits. Bad habits. Yes, no doubt they got you speaking, and learning, and did a lot of good. But now they are your enemy because you have to unlearn those terrible habits.
And what makes that task even harder is that once you reach fluency, your corrections are going to come much more rarely. People will overlook those mistakes as "cute" traits of a foreigner. "Oh I just love the way s/he talks!" So instead of correcting you, they're going to allow you to continue being wrong.
To quote Tolstoy, "Не бойся незнания, бойся ложного знания." Basically, "you shouldn't be afraid of not knowing something, you should be afraid of knowing it incorrectly."
What is your language goal?
The bottom line, then, is that whether or not you should worry about speaking correctly has a lot to do with what your intentions are with the language you are learning.
If your most important goal is to learn a language quickly, don't worry about mistakes. If you're just learning the language because you're traveling for business or pleasure, just speak and have fun. If you just want to enjoy television programs or popular songs from another language, who worry?
If your goal is to make friends and enjoy your short stay in a foreign place, by all means, take the quick gains that come from allowing yourself mistakes, because the advantages from speaking are more valuable.
However, if your language goals are more long-term, it's worth reconsidering that strategy. If you're dating (or marrying) a Spanish-speaking person, you should probably not get into the habit of sounding stupid if you want to be taken seriously. Or if you're thinking of moving to Germany, or Russia, you'll probably want to be respected at your future job. You get the idea.
Nobody is perfect, and you're going to make mistakes. If your need for the language isn't serious, go crazy with it. But if you have serious, long-term goals with the language, you probably don't want to willingly add on more mistakes to the ones you're already destined to make.