A new NBER working paper by Coibon and Gorodnichenko (C&G, ungated version here) suggests that the two approaches may be complements rather than subsitutes. The paper assumes that firms can follow one of 4 pricing schemes, a basic sticky price (Calvo) set up, a sticky information (Mankiw & Reis) regime, a full information flexible price regime and a rule of thumb regime. C&G then estimate the fraction of firms in these four regimes in the context of a more or less straightforward New Keynesian DSGE model.
C&G find that a combo of sticky price and sticky information firms in roughly equal proportions is best for matching the population moments they use to estimate their model. Rule of thumb and full information firms are estimated to be a small and not very important fraction of total firms.
This is a very cool paper. But that's not all on this front.
In a paper forthcoming in the Review of Economic Studies (ungated version here) Dupor, Kitamura, and Tsuruga consider a model where all firms face both a sticky price and sticky information scenario. They compare this dual stickiness model to both a pure sticky price and a pure sticky information model and find that data prefer the dual model. They also argue that the dual model fits the data better than a hybrid model that contains some sticky price and some rule of thumb firms.
It's like Reese's Cups people, you don't have to choose between peanut butter and chocolate!