The simple answer is yes to both. The packed sand is public. The soft sand is private. I believe the high tide line is actually considered to be public, but I just say the packed sand. If you are worried about vendors don't be, they will not bother you unless you initiate it, and then a security guard will be there to watch. If they do say something just smile and say no thanks and they will not persist.
The beach is public. You will see vendors and guest from other resorts walking by,but they have to stay at the waters edge and you can also walk the beach at the waters edge off the resort. No one will bother you unless your interested in what their selling.
The waters edge along the Seven Mile Beach is public. People can stroll up and down just like you will when you go for a walk. Security guards for CSA will keep an eye on these people and keep them moving along if needed and make sure they do not try to mingle with the guests. You will not have a problem and will enjoy the fact you can also walk from end to end on this beautiful beach. Some resorts along the north shore of Jamaica have small private beaches but they are not as nice and the sand is lower grade. You will love it here at CSA.
It is my understanding that there is no such thing as a private beach in Jamaica. CSA has an awesome beach and they do have vendors who are allowed to walk the water line. Resort security keeps a close eye on them and keeps them from hasseling the guests. You will be asked if you want to parasail, jet ski, buy crafts, have a song sung for you and "other" items. A polite no thank you moves them along.
People from other resorts and vendors can walk by. I believe vendors are not allowed to approach anyone on the beach above the water line. We stay at CN but did a day trip to CSA. CSA was not for us. Too busy on the beach and the restaurants are on either side of the resort plus CSA is the largest. We love the layout at CN. We also switch back and forth to CSS. On CN's beach, people and vendors can walk by but there is not much access. At CSA you can walk into town so they tend to have a lot more vendors on the beach. It was fun though walking down the beach to Margaritaville.