My nephew runs on Copacabana every night at midnight. From Gloria to Barra de Tijuca people run on the paths next to the beaches and nobody I know has ever had a problem. Nobody carries any money or wears jewelry other that a few reais for drinks.

Lagoa is normally croded with runners, especially on weekends, and a few people run in Jardim Botanico too. Lagoa is reportedly less safe in the area around Fontes de Saudades, but I very often go around there and I have never had a problem. People who drive expensive cars or wear jewelry in the streets next to that hill are in a riskier position, no doubt about it.