I am a US citizen, potentially to be the officer of a UK private limited company.
I will be holding 50% of shares, and another person who is a UK citizen will be holding 50%.
Do I have more tax liability for being a US citizen, vs the guy who is a UK citizen? I.e. am I systematically going to have less take home income than the UK guy, if so a 50/50 split is uneven and I need to reflect this in the agreement.
Personal taxes paid by a shareholder are a very complex issue, and are generally not thought about when structuring these types of agreements. For example: Assume you both live in the US. You have a side job that earns you $200k. The other person is otherwise unemployed. Your taxes paid on dividends received from your company might be way higher than your partner's, because other income you earned. You could say it "isn't fair" that you get less take home pay, but society and tax law says that it is fair - because taxes are owed partially based on ability to pay, and you have a greater ability to pay.
Likewise, if your partner pays more tax than you, s/he could say it "isn't fair", that they get less take-home pay simply because of where they live. But, if you are in a higher-tax jurisdiction, that (for simplicity) typically means you get more in the way of government services. For example, the US has no comprehensive healthcare system, but the UK does. Part of the cost of living in the UK might be higher taxes, but the related benefit might be healthcare.
Finally, to your specific question: Filing taxes in multiple countries is complex, but in general, for two developed countries with a good relationship [and a tax treaty], you will not end up paying double-taxes, but you might pay whichever tax rate is higher. To fully answer this question we'd need to know more about what type of income you would be taking (salary or dividends), and other specific items.
