A few weeks ago a small branch became stuck in the telephone and cable wires leading to our house. I hoped it would blow out, but the latest snow storm has put it in it's current position.
Absolutely do not try to remove this yourself. You have no way of safely disconnecting the power or being 100% sure you have protected yourself from electrocution or arc flash. The power company is typically responsible for the wire from the pole to your house. I would contact them about having it removed. I am sure they would rather remove the stick than have to replace the cable in the event it came down.
Is this a concern?
This is not an urgent or emergency issue, however it should be reported to the power company.
Could it mess up our electricity or anything?
It's very unlikely. Depending on the condition of the power lines there are a variety of very low risk events that could result from this situation. The risks are high enough to report, but this isn't an emergency situation, and no other actions beyond reporting should be taken unless you detect fire, smoke, smoldering, arcing, or anything other than a branch resting on some wires.
Once reported it is the power company's responsibility, and if any damage does occur to your structure or appliances due to the branch you may be able to hold them liable.
If so should the power company take care of it since it's before the meter?
Yes. The power companies in Ohio, and generally throughout the US, are responsible for the power lines up to and including the electrical meter. If this branch is on a powerline after your meter (say the meter is on another building and this line goes between buildings after the meter) then it would be your responsibility.
If this is the case you can hire an electrician to remove the branch. There should be a breaker between the meter and this branch sub-panel. (Typically near or in the main panel.) The electrician will turn that circuit off, verify the power is turned of in this building, and then remove the branch.
If this is on a house, it's likely that this is before the meter, and thus the responsibility of the power company.
Even if you wanted to do something, you do not have the power to do so without accepting the consequences of any subsequent problems. If you attempt to remove it yourself and damage the wires or structure in the process you may be held liable not only for the damage you cause but any subsequent work the power company has to do to restore service.
Electricians are not allowed to work on those lines unless they're working for the power company, so you can't even hire someone to take care of it for you. The lines are the legal property of the power company, and any attempt to deal with this problem could be met with legal and other consequences if discovered.