Peperonia, along with Purslane and Talinium triangulare, is one of those edible herbs that can be found growing out of the cracks of sidewalks and in abandoned niches throughout the city, few people understand that it is an excellent edible leaf, with a delicate taste, reminiscent of cilantro. It has a very shallow root and a succulent stem (also edible), it volunteers itself and grows extensively throughout my nurseries, usually self-propagating at the base of larger potted trees.
The leaves can be added to salads. In the west Indies they are used to make tea. Medicinally the leaves adn stems are used in a poultice to treat eye infections. A deconcoction of the leaves is used to lower uric acid (for rheumatism and gout).
See link for more on Peperomia medicinal applications: http://www.drugs.com/npp/peperomia-pellucida.html
The other Piper species yielding edible leaves are Piper umbellatum and Piper stylosum, there are probably others. Piper betel is also worth mentioning because, although the leaf isn't eaten as one would eat Peperomia, the leaves are used as a part of the betel quid, wrapped around the seed of Areca catechu, the betel nut palm, very common throughout southeast Asia. I am propagating the betel nut palm in my nurseries but have yet to find Piper betel.