Hello Zoé,
the key here is about the "Data entry mode" in which the chosen screening tool is configured. There are 2 possibilities: "Normal" which means that anyone's decision is immediately considered authoritative and made visible to everyone; and "Comparison" mode. In this latter case, what codes people tick will initially remain as "incomplete" coding and thus private to them.
In the Silverlight version, you change the data entry mode via right click on the root of your screening tool and "properties". In both versions, the "Normal" data entry is represented visually by the icon showing one person, the "Comparison" mode is represented by an icon with two people.
The additional facility of "auto reconcile" offered via the "priority screening" mechanism would automatically mark as "completed" (and thus public/authoritative) all the codings that satisfy the conditions as specified by the configuration in question (no auto completions, "safety first", at code level or at Include/Exclude level).
However, if you don't want to double code anything (one reference is to be screened by one reviewer), then what's needed is to configure the screening tool for "Normal" data entry (which I believe is your current situation). This means you won't be able to increase the "# of people screening each item" to more than 1 (in the "Screening" tab) and that the priority screening engine will feed references accordingly (only one person will screen a given reference). Moreover, whenever a reference will be screened, the decision will be recorded as "completed" (public and authoritative) and will thus be visible to all (all reviewers and accessible through all EPPI-Reviewer functions).
To monitor progress (by person and coding tool) you can use the "review statistics" tab in the Silverlight version (lower right corner). In the web version, the same information is visible in the "review home" tab, "Coding Progress" section (you can expand the details of each coding tool by clicking on the tool name).
I hope this helps,
Sergio