@lambad as requested in comment, here is the result of a test to show that the issue cannot be reproduced.
Added simple lightning component (API v.47) into a lightning page called "Custom Chatter Publisher".
Note that forceChatter:publisher is of context="RECORD"
<aura:component implements="flexipage:availableForAllPageTypes,force:hasRecordId" >
Assuming you are using Chatter Questions feature, then editing feed posts (the questions) is not permitted for Community users with some exceptions:
For community users, feed post editing is available only if the community was created using either Salesforce Tabs + Visualforce or the Customer Service template.
Given you meet the above, user's need the ...