Ecuador does accept dual nationality - the important thing is to find out whether your birth country does too. Bear in mind however that the process is time consuming and costly.
In order to start the process of getting citizenship, you either need Ecuadorian parents (consanguinidad) or to have become naturalised. In order to become naturalised, you need to have been living continually in Ecuador for three years.
Like the visa registration process, you need to go to Quito to carry out the procedure. As you’d expect, there’s a lot of red tape involved, plus some very high fees (it will cost over $1,000 to become a fully naturalised Ecuadorian and can take up to 7 months). Bearing all this in mind, you might want to consider just living in Ecuador on a resident visa if you’re planning on staying long-term.