First we need to take into account that com badges are not just communication devices. They are not simple phone analogies. They allow (probably encrypted) subspace communication over 40000 km, are used as transporter location and boosting beacons, allow remote access to ship computers, have embedded universal translators, etc. They are highly advanced and important technology. These are akin to current day encrypted military satellite phones.
They also signify Starfleet commission. They are badges in the same way modern day police have badges. They can denote rank and position. So non starfleet officers (i.e. personnel) will not have them.
The Enterprise does have an internal communication system as well. This is accessed by general computer terminals. Communication over this can be secured or unsecured, and the com badges can be linked into it. As we rarely get glimpse of civilians at work or personal in the show, we can only make general assumptions. The Enterprise is a mixed population ship, fairly large, so any design would recognize the need to have some capacity for civilian inter ship and ship to planet communication. DS9 shows Jake and Nog can communicate and receive personal live communications. Voyager shows that personal communication is restricted when bandwidth was limited.
But in TNG two instances of Civilian to Officer communication does happen over com badges, for personal matters. Both involve the on board school. Worf and Data are notified that their kids aren't fitting in and that they need to go pick then up.
essentially, civilians use the same ship communication system, but just don't activate it through the com badges they lack.