In this paper we propose to study the dynamics of financial contagion between the credit default swap (CDS) and the sovereign bond markets through effective transfer entropy, a model-free methodology which enables to overcome the required hypotheses of classical price discovery measures in the statistical and econometric literature, without being restricted to linear dynamics. By means of effective transfer entropy we correct for small sample biases which affect the traditional Shannon transfer entropy, as well as we are able to conduct inference on the estimated directional information flows. In our empirical application, we analyze the CDS and bond market data for eight countries of the European Union, and aim to discover which of the two assets is faster at incorporating the information on the credit risk of the underlying sovereign. Our results show a clear and statistically significant prominence of the bond market for pricing the sovereign credit risk, especially during the crisis period. During the post-crisis period, instead, a few countries behave dissimilarly from the others, in particular Spain and the Netherlands.