How should such an issue be addressed, so that it doesn't become a
problem and impact productivity and well being of the staff?
I've been through this process many times - as part of the acquiring company, and as part of the acquired company.
It's a real challenge to keep people focused on their day-to-day work, when they know that a potentially disruptive change is coming their way. It's natural for people to want to know "how will this affect me?" and to fear the worst.
Since you already chose not to keep this secret, you must now address this issue quickly and directly.
Gather your staff for a frank discussion. Be as transparent as you can (within the bounds of your business needs). Tell them what you know. Let them ask questions.
Try very hard not to over-promise. Unless you have already formally made such a decision, do not tell your staff that all of their jobs are safe. I went through one merger where we were told not to worry about our jobs, only to have some folks laid off within a month. The rest of us who remained could only conclude that our management lied to us, and I never trusted them again.
Instead, let them know that you'll be evaluating the newly merged staff as quickly as you can, and will give people as much notice as possible. If you already know the process you'll use to do that evaluation, you might choose to share that process with the team.
Give your staff a feel for your vision for the merged company - what you think it can do that the two individual companies couldn't, why it's a good thing for the company, and where it's a good thing for them.
Acknowledge their unease (don't just say "don't worry"), and assure them that you are in this with them.
In my experience, mergers/acquisitions work best when the duration of unsettlement is as short as possible. While always painful, a company that has one quick layoff after a merger is a far better place to work than a company that has several layoffs stretching out over time. The uncertainty was the most productivity-killing part. Get it behind people as quickly as you can, then tell the team that "the painful part is over - now we can grow as a new [merged] company".