I have heard the argument made that in LibPar, the services made by government employees would be filled by the FreeMarket™.  I wholeheartedly agree with that position.  The reason why I believe that, is that I believe people are generally good and would love to help each other...when not prevented by the government (see cases like where senior citizen went to jail for feeding homeless).
    But why wait for LibPar?  There actually are real world cases right now.  And those cases of non-government oversight is fairly common.  (I can already see smoke coming from the Statists' brains as they say rooooaaadds).  If you look at these instances, you can understand what LibPar might look like based on how things really are now.
    1. Water and Sewer.  I have a private well and my own septic tank.  This system requires little maintenance and keeps our property self-sufficient.  Everyone in our area has their own well and septic.  Pretty common, actually.  No water department over here.  In LibPar?  No change.
    2. Fire Department.  We have a 100% volunteer fire department here.  The fire trucks, equipment and building are paid for with county taxes (property), and the firemen receive a $1,000 per year stipend for maintaining their equipment (boots, outfits, uniforms, gloves, etc).  These guys have full time jobs, but answer the call when the radios go off.  In LibPar, financing would be through fundraisers (hog roasts, fish fry, donations, etc). Small change.
    3. Schooling.  Right now, there are plenty of alternatives to public school (or charter schools funded by tax dollars).  Homeschooling, unschooling, private schooling are all valid options that operate as they would in LibPar.
    4. Police Department.  This is where LibPar usually gets a great disconnect to separate current standards from paradise.  With any instance of police corruption or violence, LibPar advocates say "private security would replace that".  That is true..to an extent.  

     But let me tickle your brain here for a moment.  What about a voluntary police force.  You know, like the volunteer fire department.  Instead of paying police a salary, they would receive a $1,000 year stipend to pay for maintaining their uniform (boots, costume, etc).  Like the voluntary fire department, you would need to provide equipment (tazers, cars, radios, vests, etc).
      Do you think it would work?  Well, of course it would.  Because it already does.  That exact scenario already plays out in a number of communities as it has for many years.

     In fact, I  interviewed with such a department for a 'reserve' position.  In this small town, there are more volunteers than paid staff.  The reserve officers are on call, but also run regular scheduled shifts.  They have the same uniforms and 'authority' of the paid officers.

     As you can imagine, the officers are more involved with the community and maintain positive relations.  The Town Marshall (equivalent of police chief) assured me that all of the officers issue tickets..at their own discretion.  They do not maintain quotas of any kind.

    They still deal with unpleasantries, like: domestic abuse, trespassing, theft, and car accidents.
    But in LibPar, I would envision the LEO to be more like this than private-contractor-security.  So are we closer to Libertarian Paradise than we thought?