As @lbf notes, the "either/or" construct is good, though honestly the original statement as written in the OP is not at all impolite.
However, as English is commonly spoken and understood by a vast number of folk, there may be contextual clues that could help. Are these foreign engineers all from the same country? If so, where?
I'm making some assumption here:
- You are Japanese, as is your company, and you are hosting engineers from abroad.
- Those engineers are from the same predominantly English-speaking country
If the above are true, consider the following (though there are variations even within these samples, so... take that as you will).
If they are American, the either/or construct is fairly commonplace. It's not exactly overly friendly, but it's concise and cordial. Midwestern Americans and Brits may, however, feel a sense of obligation to attend one of the small parties rather than dine by themselves. It's an oddity of politeness, and there's not much that can be done via a single e-mail to truly instill that either is absolutely as good as the other.
Australians and Southwestern Americans are generally more likely to accept the either/or as truly being equal. Both tend to have a more sane and settled view of social politeness (where a goodbye can actually just be a goodbye).
For many folk, "or you can have dinner by yourself" may sound like a "well, or you can be anti-social and do your own thing", which is part of my above comment that they may feel obligated to attend one of the arranged dinners. I believe that's not at all what you mean, of course, but again... oddity of politeness (on their part, not yours).
The most neutral way you could write it, in my experience, would be to put the latter first; something such as:
There are two options for dinner. If you'd like, you may explore and dine where you'd like (there are many great options in Tokyo an Osaka), or you can have dinner with us at a pre-arranged restaurant in either location. Please RSVP with your preference so that we can plan properly.
This has the benefit of saying up front that their own dining experience is an option, as well as giving them the out (exploring the area). Then you let them know that there's another option (maybe they aren't quite so adventurous?) and that you just need a head count either way so that plans can be made. This takes most of the pressure off for some folk. It may seem silly, but it definitely rings true for some.
I can't speak to other countries, but adding that detail to your original post may help other answers if they aren't covered here.
Source: Me, a Midwestern American who has traveled a lot and spent a good bit of time with Brits and Aussies in both professional and military capacities.
