The Local Completion Mode works with proposals you have applied locally and does not download or upload any data. Your applied proposals are not persisted, and are only used to recommend proposals during the same Eclipse session. This provides you with realtime feedback, where each applied proposal is immediately visible in the next proposal window.
The Shared Completion Mode downloads recommendation models that are based on the applied proposals of all users of a particular server. Your own applied proposals are uploaded and integrated into these models. This strategy does not give you realtime feedback, but the ordering of proposals will change after a model has been updated.
Local + Shared Completion Mode combines the realtime feedback of the Local Mode while also dowloading models. Proposals are ranked according to both the shared models and your locally applied proposals. As with the Shared Completion mode, your applied proposals are uploaded to the server to be combined with all users' selections.