Protein deficiency in vegetarian diets is a long-time myth that is unfortunately still quite prevalent. A couple of years ago I would've asked the same question.
It turns out that most people get much more protein than they need, and this extra protein, especially from animal sources, is harmful to long-term health. Something that many people have noticed is that a diet rich in plant-based whole foods helps in maintaining a healthy weight. It is certainly possible, however, to malnourish yourself by not having enough variety in your diet. The myth that you need to compose each meal so as to provide complete proteins has been thoroughly debunked, but that, of course, doesn't mean you can survive forever eating just celery and carrots.
Lacto-vegetarians tend to get tons and tons of protein from dairy products. Mind you, this is horrible, cancer-promoting protein, but complete protein nonetheless.
Many people who eat a strictly plants-based diet find that they consume a much wider variety of foods than they did before. Eating more whole grains, nuts, and legumes is a healthy way to get protein in your diet.
-Brian
BBC article on vegan protein sources Wikipedia article on protein in nutrition