I received a common question: "How can my students collaborate on a map, like they do in an online doc?" With so many schools using Googledocs and similar tools, this is understandably confusing. Student1 makes a map, saves it, and shares it in the Org/Group. Student2 opens it, makes changes, and wants to save those back. Generally, no, that won't work. ArcGIS Online is not architected like Googledocs, capturing every individual tweak anyone makes. In ArcGIS Online, saving only happens when you specifically choose to save, and saves within the user's account, and saves only the current contents and configuration in that browser.
Here's an example of a single student signed in via two browsers at once on one computer:
Tommy opens his map in two different browsers (1 Firefox, 1 Chrome) like this:
[] Tommy opens the map in Firefox, makes 4 changes, but doesn't save yet.
[] Tommy opens the same map in Chrome, makes 3 different changes, and does save#1
[] Tommy makes 2 more changes in Firefox, does save#2, and exits.
[] Tommy makes 1 more changes in Chrome, does save#3, and exits.
The final saved map will have only the 4 changes made using Chrome, because it opened the original condition and saved with a total of 4 Chrome-based changes. Whatever was saved by Firefox during save#2 was overwritten by the changes from the final situation in Chrome, completed in save#3.
This represents what would happen in an Org with a single user in two different browser windows, or with an Org-based user who shared login/password with another person (which is a no-no), or with a "shared public account" (sharing logins is permitted for edu/NPO/NGO uses in public accounts, but it can be very hazardous for this reason and others; see http://esri.box.com/agousestrategies).
SUMMARY (assuming "standard ArcGIS Online operation" as found in typical school use):
Sharing a map with a group/Org lets other users see and use and "SaveAs", but they save copies in their own account
You CAN have multiple people adding features into a single editable feature service at one time.
You CAN have people creating individual layers and sharing those layers with each other.
You CAN have people working on individual maps which subsequently get collected into a single story map.
BUT only ONE user, in ONE browser, should edit one map or app at a time.