No third country can tell another whom they can sign trade agreements with. So China can sign "trade agreements with low standards for labor, the environment, food safety, etc., with other nations" all they want and undoubtedly will.
If we are going to get away from agreements with 'low standards' like the ones we have now and China will continue to pursue, how do we do it? FDR and Truman tried to do it with the ITO which was a 'high standards' trade agreement (that congress shot down in the name of national sovereignty). There were 22 nations that had signed as members. The idea was to tie trade to labor standards, business regulations and full employment goals.
FDR and Truman understood that the countries involved would be trade with other countries that were not members of the "high standards" ITO but they pushed it anyway. They believed that the way to tie 'high standards' to trade was to negotiate multilateral agreements that specifically did just that.
IMHO, despite the fact that China can sign as many "low standard" trade agreements as it wants (and it has been doing pretty darn well just using WTO rules for that matter) is no reason for others to avoid negotiating "high standards" agreements (if that is what TPP is).
Will the TPP actually force Vietnam to raise standards across the board?
In other words, by passing the TPP will we really be writing the rules?
Good questions. If any international agreement is to be effective it has to have enforcement mechanism. You cannot, of course, just rely on countries (including the US) to do what they agree to do. Enforcement mechanisms will have to have 'carrots and sticks' to insure compliance.
I don't see it as a case of "we will be writing the rules" though that is how Obama stated it. As with the ITO, the "rules" are the result of negotiations with many countries. They are not exactly 'our' rules. If they have high standards that really won't matter to most of us. (Heck, our labor laws could use some 'standard-raising'.) But if the 'rules' as written in TPP are 'high' and 'enforceable' then it's a good thing. The reason congress rejected the ITO was that corporations saw that they would have to follow 'high' standards with international enforcement. They didn't like that and lobbied congress to kill it. Better for them to keep enforcement supervision questions in the hands of a congress which they have a lot of direct 'influence' over.