To fully characterize the information that two `source' variables carry about a third `target' variable, one must decompose the total information into redundant, unique and synergistic components, i.e. obtain a partial information decomposition (PID). However Shannon's theory of information does not provide formulae to fully determine these quantities. Several recent studies have begun addressing this. Some possible definitions for PID quantities have been proposed, and some analyses have been carried out on systems composed of discrete variables. Here we present the first in-depth analysis of PIDs on Gaussian systems, both static and dynamical. We show that, for a broad class of Gaussian systems, previously proposed PID formulae imply that: (i) redundancy reduces to the minimum information provided by either source variable, and hence is independent of correlation between sources; (ii) synergy is the extra information contributed by the weaker source when the stronger source is known, and can either increase or decrease with correlation between sources. We find that Gaussian systems frequently exhibit net synergy, i.e. the information carried jointly by both sources is greater than the sum of informations carried by each source individually. Drawing from several explicit examples, we discuss the implications of these findings for measures of information transfer and information-based measures of complexity, both generally and within a neuroscience setting. Importantly, by providing independent formulae for synergy and redundancy applicable to continuous time-series data, we open up a new approach to characterizing and quantifying information sharing amongst complex system variables.