In light of offshore wind expansions in the North and Baltic Seas in Europe, further ideas on using offshore space for renewable-based energy generation have evolved. One of the concepts is that of energy islands, which entails the placement of energy conversion and storage equipment near offshore wind farms. Offshore placement of electrolysers will cause interdependence between the availability of electricity for hydrogen production and for power transmission to shore. This paper investigates the trade-offs between integrating energy islands via electricity versus hydrogen infrastructure. We set up a combined capacity expansion and electricity dispatch model to assess the role of electrolysers and electricity cables given the availability of renewable energy from the islands. We find that the electricity system benefits more from connecting close-to-shore wind farms via power cables. In turn, electrolysis is more valuable for far-away energy islands as it avoids expensive long-distance cable infrastructure. We also find that capacity investment in electrolysers is sensitive to hydrogen prices but less to carbon prices. The onshore network and congestion caused by increased activity close to shore influence the sizing and siting of electrolysers.