A board filled with sticky notes (or a Jira board) is as simple as you can get. A glimpse on the board should give enough information about the progress of the team, and if it doesn't, it means that the board is wrong and should be corrected.
A real, physical board is usually better, because:
It is more visual than any software ticketing tool I've seen so far. Don't get me wrong: I use Jira on daily basis and I enjoy using it, but I still find that in terms of progress visualization, the real board is more capable.
Any person can see your progress just by physically passing through your office space. One doesn't need to start a PC, go find the link to Jira's board (one of the links everyone outside your team always loses), authenticate, etc. Moreover, being physically where the team is, the person may even ask questions if the physical board is not clear. “Hey, you told me yesterday that you finished changing the interfaces used by the ETL; so why is the task in progress?”
Nevertheless, would it be a real board or a virtual one, it should allow someone outside the team to understand:
The current progress, that is:
What you have done during this sprint,
What are you working on right now,
What you still have to do for this sprint,
The priorities,
The blocking points (bonus points if the reason of blocking is visible on the board).
Not only should it be able to communicate those points, but it should also be able to show it precisely. What often happens, especially when using virtual boards, is that tickets grow, and grow, and grow, and someone can easily spend five days working on a single ticket. This, by definition, makes it impossible to visualize progress. You could, of course, ask the developer what is his progress on a given task, but the answer would be irrelevant (see, for instance, the ninety-ninety rule).
So keep every task small enough. If a task takes two hours, that's great. If it takes one day, it's a very, very large task. If it takes several days, you have to split it into smaller tasks, in order for the board to reflect the progress of your team.
Once you do have tasks which are granular enough, “communicat[ing] the integration readiness” of a task becomes easy: either the task is done, or it's not. There is no 42% done or 85% done.
A few additional pieces of advice:
If integration is painful, it usually means that the interfaces weren't designed carefully enough. Sloppy work at this level results in hours, days or months of wasted time for both teams; do spend enough time designing the interfaces between your team and the outside world.
When it's difficult to design an interface or when it should be changed while both teams already started working on their respective parts of the code, do work together with other teams. Don't just do phone calls. Go see them, or invite them to come and work with you, side by side. I stopped counting the number of hours, days and months wasted by teams where members are just too lazy to walk to the other side of the office building, and decide to communicate by e-mail.