We have created sample project and there are five Issue Types: Story, Epic, Task, Change Request and Bug for that particular project.
Not out of the box. JIRA handles access at the project level. The behaviors plugin might do it, but I don't know if it is available in the cloud. I suggest you create a project for the different issue types that need to be segregated. You can link across projects with the right permissions.
Take a look at the issue security schemes: https://confluence.atlassian.com/display/JIRA/Configuring+Issue-level+Security
You can set the issue security level in workflow post function, differently for each issue type.
Having said that, creating different projects might be better, as Joe already suggested!
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