There are many immersive living Latin programs in the United States and Europe. I am not sure about elsewhere. These communities of teachers and learners do form places where you can experience Latin as a modern spoken language in the way that seems to interest you.
The annual Living Latin in NYC conference is an example of a weekend gathering with lectures, classes and socializing all done in Latin. You can even take a class in spoken ancient Greek.
Here's a lecture in Latin De iucunditate et utilitate Auli Gelii Noctium Atticarum from that conference to get a sense of what it's like to have Latin in your ear.
In addition to this, there are other organizations promoting similar experiences of speaking Latin as a modern language. The North American listings for such events and programs can be found at the North American Institute of Living Latin website In Europe Luigi Miraglia is well known for his programs at the Accademia Vivarum Novum in Italy, including an international summer school. Reginald Foster is well known as a major influence of many of these experts. He teaches a summer program in Milwaukee.
Perhaps, others will add to my answer with knowledge of these communities of Latin speakers who gather in various ways and times to learn and experience the Latin language in this deep and lively way.