I've yet to shadow a DO, but I'll explain my experience with a chiropractor.
Getting an LOR is like trying to land a date with an acquietance. Always put your best foot forward...it helps if you're naturally a ppl/happy go lucky person. If not, become one. Spend enough time with the DO until you feel they have seen the best of you. From my experience, this can take as little as 4-5 exposures.
Your initial visits with your LOR writer should demonstrate your passion, inquisitiveness in the profession as a whole but also in what that specific DO does (how long practicing, motivations, philosophy, etc). Hopefully, this part comes naturally as I wouldn't recommend faking interest. Now that you've laid the groundwork, start talking about yourself more (interests in the profession, what you're studying, any acheivements big or small, etc). You have to make it seem (again, helps if it comes naturally) as though your shadowing is contributing to your desire to become a DO. It most likely will anyway (I hope). When you've become comfortable enough with your referee that your level of communication is more along the lines of a favourite uncle/aunt (very intimate but still respectful), that is the time to ask for an LOR. It's impossible to say when that time will come, but be patient. Of course, you can always get a letter before that point and a good one at that, but it will never be as personalized.
That's been my experience so do take my 'advice' with a hulk-like, huge a*s grain of salt.
Best of luck and I urge you to see the experience as an opportunity to become a better/more informed person and not just a LOR factory.