We have been together for 6 months. We are far away. We meet on the weekends. I have been to paying, but now I think it's time to share. We also talk about living together. How can I discuss this topic without being impolite?
I'm from Europe and not a native speaker, so please take this into consideration.
See it from her perspective. You started dating and you paid for everything. Now that she is officially your girlfriend, you stop paying or at least are not as generous anymore. I'm totally fine with splitting bills. But changing generosity after the dating phase can easily being mistaken as you feeling to save and slacking. Try to start off your living together with a bank account or jar or whatever where both of you put an equal amount of money in every month. This money you take for rent, food, going out, and so on. So the question is answered straight away. Everybody pays equal parts.
I totally understand you wanting to split bills. It should have been done from the get go and not after half a year of dating.
How to ask her to pay without leaving a strange feeling on her side: Start a conversation about cost sharing in the future. See what her opinions are. If she is totally for sharing costs, then maybe that is a good time to ask her, if she wants to pay the next time you go out. Or discuss the shared account in the future for all your joint exspenditures.
Edit for clarification: This might work for several reasons. First, it doesn't come out of nowhere as you are already talking about sharing exspenditures. Second, OP will know his girlfriends opinion on the topic. If she is positive about sharing, he can mention, he likes to share costs for going out in the future. If she is not positive about sharing, he will have to discuss this first. They want to move in together. Financial issues should be discussed beforehand anyway. Maybe she wants to be courted when going out and is fine with spending more money on the groceries in return. So this could be a solution. Talking about the topic in general and using this disussion to make clear, who wants what in the realtionship is in my opinion better then just making her pay there and then in a restaurant out of nowhere after half a year.
I hope it is more clear now. I know it is written a bit harsh but I'm trying my best with the english language.