As a convert to Catholicism from the Evangelical community, I just want to clarify your misunderstanding of these phrases.
To say that you have a personal relationship with Christ, does not negate your corporate relationship, it actually means you have taken it a step further and have gotten very serious about knowing Christ, emmulating Him and allowing Him to do His will in your life. It’s like the difference between knowing about someone and being loyal and faithful to him, and getting to know that same someone intimately, personally.
And of course, this all happens through the reading of Scripture, through prayer, and is all accomplished through the workings of the Holy Spirit. The one advantage that the Catholic has is the addition of the Sacraments in this process. It’s just very unfortunate that many Catholics do not take advantage of the Grace that is available to them to come to a more committed, intimate relationship with Christ through all the above means.
I don’t think that any Catholic who is serious about his faith should be in anyway threatened by the question of personal relationship, because the answer would certainly be “YES!”.