What does 一起 imply about a romantic relationship?
在一起 (be together) is a subtle way to say "做情侶" (be lovers/ a couple) 

be boyfriend and girlfriend, common law partners or husband and wife

just like 喜歡 (like) is a more subtle way to say 愛 (love) 

Most Chinese are quite reserved, words like 做情侶 (be lovers) or 愛(love) are rarely used directly