Yes they can.
A nice benefit of this approach is that, since most reports are assessment-specific, those who can see data across the goals will be able to see summary-level data across the goals.
It is important to understand that students can only work on the same assessment in one goal. This is useful if you have multiple instructors using the same assessment - students will easily be able to move between the instructors' goals, for example if their class schedule changes. (You can find more details about how that works from a student perspective here.)
If you're expecting students to work in the same assessment (same scope + target score), but want them to start from scratch, you have to use a different assessment. For example, you may have High School Juniors working on ACT Prep, and then they'll study for the test again in their Senior year. If you use the same assessment with your students (even in new goals), they will pick up where they left off. That means your Seniors won't start the year by taking the initial diagnostic, but instead will continue with the progress made as a Junior. If you want your Seniors to start over with a blank slate, you would need to create a new assessment for them to work in. It can be identical (except for the name) to the one they used previously... in fact, you can clone the original assessment to ensure it has all the same properties.