I thought I would add a late answer here to expand upon my comments from last year.
Another example using Open Source software (PostGres/PostGIS, GeoServer) is the TreeKeeper suite from Davey Resource Group (Davey Tree Expert Company). Since I currently work there, I can give a little more information on how things operate. The info is stored in the PostGres/PostGIS backend and fed through GeoServer to OpenLayers. The OpenLayers site is hosted through a ColdFusion server although it could as easily be done through PHP or other server type. Although not shown in the demo, the field personnel, clients, and others involved in the projects are able to create, update, and delete features via the web interface and there is a mobile interface, too.
I was not involved in either of those two projects directly. However, I am familiar with the way they operate. For the TreeKeeper site... The user sees a page rendered in OpenLayers/HTML. Whenever they edit/create/delete, that request is filtered through an AJAX (via jQuery or other JS library) request to an API set up in ColdFusion that then runs a query to the PostGres/PostGIS database server making the appropriate changes. Changes then spread out to GeoServer to adjust tiling, if necessary, and other mapping related items. Once the changes are done, the webpage will either reload (saving its state) or respond to a AJAX response change to update the view for the user. I am assuming the City of Hudson's sites operate in a similar fashion, but I know they use different APIs for their work.
Basically there is a chaining of requests/events that happens to make sure things are updated accordingly.
There are numerous projects out there to support this. I keep seeing new web APIs pop up, server side technologies, hosting services, etc etc every time I Google web mapping. Once you know your set up and the availability of technology (not to mention the rogramming/server skills of your staff), you can design a workflow that will send information back and forth to users. If you're limited on knowledge and have a flexible budget, ArcGIS Online or other subscription based service may be the easiest for you since they usually handle all the heavy lifting of data storage, request routing, authentication, and user interfaces.