Like others said, give a short,concise, and a partinent turn over. If you're rambling about stupid stuff the medic is just going to ignore you anyway and start assesment over, and if you keep rambling it just takes that much longer for him to start doing his whole assesment again.
I advise the OP to sick with gender, age, mental status, C/C, V/S, and demographics if you got that far. Example, the medics walk in, you say: female, 66, verbally responsive, oriented to name only, family advises her BGL is 40, BP 156/92, P 92, RR 20, we're getting their name, SS#, med list, etc. That takes all of 30 seconds. That's actually how I begin my face to face report to the triage nurse. They want a long winded story from EMS as much as medics want a long winded story from EMT's.
When you get onscene, say hello to them from across the room, and see how they respond. Go up to them, and slowly grab their wrist for a quick pulse check while you speak to them. Now you have the start of a mental status eval, a pulse, skin CTC, and how they're breathing (are they talking in complete sentences, etc). Have someone knock out a BP and resp count while you determine a chief complaint, and get a third provider to secure meds and demographics, if available. Meanwhile, you can open up the stair chair, lay out a sheet, place them on the chair, and put them on O2. That takes all of three to five minutes. Now you look like superstars to the medics. If the medics had a greater than 10 min ETA initially, if the pt and their condition can tolerate a chair, then you should have them downstairs or on the way down (radio this to the medics when you leave the apt), The medics will heart you big time if you can do that.
Occasionally the medics will radio for pt info as they arrive onscene, before they come inside. Now you have something to say other than "we just got here, we haven't had a chance to do anything yet".