Must consumers using 10 10 long distance services also have at&t or
other phone companies services?

http://www.puco.ohio.gov/PUCO/Consumer/information.cfm?doc_id=1186

You just need to enter a 3-digit access number after 1010 to get hold
of a long distance carrier after which it would automatically dial the
destination number. I do not think you need any connection with any
other service provide.

You do not need a long distance provider.  Most local companies when
you do not have a long distance provider put a block on long distance
calls to prevent a call going through and you getting charged "random
rates".  Random rates aren't how they sound, it's upwards of 2.00 a
minute depending on your local carrier.
So, just contact your local carrier, tell them you don't want a long
distance carrier, but you want the ability to be able to dail long
distance numbers, but won't be dailing them directly.

Use phone cards from online stores. I call Russia for 3 cents a
minute, USA (long distance) - 1 cent a minute.

To mscorleone: Your reply was just the answer I needed. Thanks.