Yes, there are Boolean monotone functions where the smallest monotone circuit is exponentially larger than the smallest circuit that uses NOT gates. However, the majority function does not have that property.

Here's how to build a polynomial-size monotone circuit for the majority function. First, build a box with two inputs and two outputs, where one output is the AND of the inputs, and the second output is the OR of the inputs. Clearly, this is a monotone circuit, since it doesn't have any NOT gates. But it is also a 2-input sorting function. If you send in two bits, the OR output will be the maximum of those two bits, and the AND output will be the minimum of those two inputs. (Just check all 4 cases, if it isn't obvious that that is true).

Now look up sorting circuits in Knuth. He shows a huge number of different ways of sorting N inputs using only a polynomial number of 2-input sorting functions. Since his circuits use nothing but 2-input sorting functions, and since each of them is a monotone circuit, the entire thing will be a monotone circuit.

Now it's easy to build a majority circuit. Given N input bits, use Knuth's circuits to sort them. Then output the middle bit of the sorted list. That's the majority.

The first known example of a monotone function whose circuit reduces from exponential size to polynomial size by using NOT gates was more complicated than a simple majority function. It was something about finding perfect matchings on a bipartite graph, if I remember correctly. It wasn't terribly simple. I don't know of any examples that are as simple as a majority function.