In regards to AOA, if you are doing ACGME residency and no need for Resolution 42, then there is no obligation to remain an AOA member
If you plan to obtain Resolution 42 (for the AOA to recognize your ACGME PGY1 year as an AOA year), then you will need to remain a member of the AOA at least during your intern year. If you want to take the AOA specialty board (ie be board certified via AOA pathway instead of ABMS), then you will need to remain a member for Resolution 56. This is usually done by people who did an ACGME residency but accepts an AOA fellowship (and wants to be board certified in that specialty) - such as ACGME IM residency followed by an AOA GI fellowship, then taking the AOA IM and GI boards. Currently there is no way to take the ABIM GI boards if you do an AOA fellowship. (*in the future, all residency/fellowship will be acgme so this may be moot)
Otherwise you may need attend occasional AOA conferences for AOA CME credits IF your state requires AOA CMEs for licensure or renewing licensure. However you don't need to be a member to attend (you'll just pay the higher non-member price)
The AMA is more of a political advocacy organization and doesn't have the same "power" over MDs that the AOA have over DOs. It is believed that currently only 11-15% of active physicians (excluding medical students/residents) are active due-paying members of the AMA. That number was closer to 75% back in the 1950s. A recent survey showed that a majority of practicing physicians do not believe the AMA represents their views or advocate on their behalf. It would be interesting to see what the percentage would be if the AOA ask DOs (all DOs, not just AOA members)
The specialty colleges (american college of physicians, american academy of pediatrics, american college of surgeons, etc) are where most members may hold memberships or attend conferences, BUT membership isn't require to maintain ABMS board certification. You do not need to be a member of ACP to be ABIM certified, nor a member of AAP to be a board certified in pediatrics by ABP. You do have to maintain membership if you want to maintain and use those fellowship postnominal designations (ie FACP for ACP, FAAP for AAP, FACS for ACS, etc)
As you can see, on the MD side things are more "fractured" and not consolidated as it is on the DO side. As you progress through your training and career, more and more organizations will try to entice you to join (ie pay dues) .... everyone wants your money, even your medical school and undergrad will ask for money (while you pay off your student loans)
Refernces:
http://www.physiciansweekly.com/is-the-ama-really-the-voice-of-physicians-in-the-us/
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3153537/
http://www.modernhealthcare.com/article/20130509/NEWS/305099950
