Now, I wasn't old enough by then to remember such details from that time, so I really can't tell. But I'm sure I had never left Estonian mainland before the reacquisition of independence.
As much (or as little) as I asked around, nobody remembered exactly. Although, the general consensus was that things were probably still approximately the same as before.
From the stories that I have heard, it was indeed somewhat possible to visit foreign countries (although I'm not sure starting from when exactly). There was some concept of tourism trips to "selected countries" in "supervised" groups, which meant that you could get this opportunity through your workplace or some other organization that you were part of and had to apply for some preselected destination. If you were deemed to be "safe" enough to be let out of the country (and to this specific destination), you got the permission.
Secondly, if you had some relative in another country and he/she sent you an invitation, then based on that invitation you had the opportunity to apply for a permission. Of course your "safeness" and whether you had a reason to return, was also taken into consideration when deciding whether to give you the permission or not.
The same applied to when you wanted to travel to Estonian islands, and even getting to mainland coastal areas was difficult or even impossible for non-locals. After all, the Soviet Union couldn't allow people to get to places where it's easier to flee the country.