If you are competing for resources, it is because you have not agreed with the other project lead what the split of resources should be. Your project documentation should define your resource requirements, including the finances, the equipment, and the people. If that isn't the case, then you have a hole in your documentation, so a first step might be to fix that. This should then allow you to have a sensible discussion with your colleague, and possibly with your manager / his or her manager / your project sponsor / etc., so that you can reach a position where you know how the resources should be split - and also so that you can manage your finances. If your people are being pulled onto the other project, you must (presumably) be underspending on people costs, and this should show up in any project reviews - again allowing you to have a discussion about what is needed.
Your organisation will likely have some sort of priority between the different projects, so maybe it is happy for you to lose people to allow the other project to progress faster. That's all fine, but you should know that. Or maybe it wants your project to take priority, in which case you should be able to tell your colleague this, and that person then has to make a case to make a grab for "your" people.
Ultimately it comes down to managing expectations - yours, your colleague's, and the organisation's. If you are responsible for delivery and you have not delivered because you gave away your resources, you need to be able to justify that and defend your decision, or face the consequences. It's great to be nice, but sometimes you also have to be strong and hold your ground, unless told otherwise by someone who has the necessary authority. And just remember that if your manager says one thing and the other person's manager says something else, escalate the issue, provide the ammunition, and let the two managers fight it out.