This article considers sample size determination for jointly testing a cause-specific hazard and the any-cause hazard for competing risks data. The cause-specific hazard and the any-cause hazard jointly characterize important study endpoints such as the disease-specific survival and overall survival, which are commonly used as co-primary endpoints in clinical trials. Specifically, we derive sample size calculation methods for two-group comparisons based on an asymptotic chi-square joint test and a maximum joint test of the aforementioned quantities, taking into account of censoring due to lost to follow-up as well as staggered entry and administrative censoring. Our simulations demonstrate that the proposed methods can produce substantial sample size savings over the classical Bonferroni adjustment method and generally have satisfactory finite sample performance. We illustrate the application of the proposed methods using the 4-D (Die Deutsche Diabetes Dialyse Studie) clinical trial.