As someone who did the same as you did, you only need proof of insurance for the first trip (the trip whose details you provide as part of your visa application).
The insurance must have a minimum 30,000 EUR policy and cover the dates of your trip. I have used https://www.axa-schengen.com/en and found it to be easy and quick.
Assuming you get your multiple entry visa, for your second trip - you will need to obtain insurance again and carry proof of the same with you (along with other documents pertaining to the nature of your visit). You won't be visiting the embassy (since you already have a visa) but the immigration officer may ask for further documentation regarding the nature and purpose of your visit.
To be honest, I have yet to be asked for proof of medical insurance; I have been asked for hotel/ticket/purpose of visit etc. questions, but never "please show me proof of medical insurance"; however as the cost is trivial compared to the cost of the overall trip, it doesn't hurt to keep the paper with you; and hope you never have to need it during your trip :)