Please check the below steps which explains my issue.

Umbraco version:  6.1.5

Steps to replicate:

Login to Umbraco as "Admin" user

Adding New DataType:

• Go to Developer node
• Add a new Data type called "UserGrid" using "UComponents: DataTypeGrid"
• Add Data Type "TextString" into "UserGrid" as Image description 
• Add Data Type "Media Picker" into "UserGrid" as Product Image and Save it.

Adding New DocumentType

• Go to Settings node
• Create a new Document Type "Showrooms"
• Go to "Tabs" of the Document Type
• Add a new tab "About us" into document type "Showrooms"
• Add another one new tab "Products" into same document type
• Go to Generic properties 
• Add New Properties with "Simple editor" on "About us" tab
• Add new properties with "User Grid" (Custom component created with UComponent grid) on "Products" tab and Save the data type.

Adding new Content 

• Go to content node.
• Create a new content "Showroom1" with the document type "Showrooms"
• Open the new content "Showroom1" and add value on "About Us" and "Products" and save it.

Create new user:

• Create a new user "Provider" with User type "Writer"
• Give the permission to access the content node showroom and save it.
• Log out from the "Admin".

Issue to replicate:

• Login with new user "Provider" 
• Go to Content node
• Click Showroom content. 
• Go to "Products" tab and Click Edit on an item.
• Pop up comes with saved values. (Issue 1: The tab is getting redirected to "About Us" tab. Not placed on "Products" tab which has the grid)
• Modified the content and click save. (Issue 2: Save button triggered with out any issues. But the modified content is not getting saved.

*** This issue only comes on "Writer" type user. If I login as "Admin" and modify the content, then It gets saved perfectly with out any issues.