How do I know if my propensiy score estimation is correct? Are there any test statistics, visualization methods, or other methods to help me decide whether my propensity score estimates are correct?
 A: It's impossible to know if any estimate is correct, as AdamO mentions. You can assess whether your estimated propensity scores are adequate for reducing bias in an effect estimate by assessing covariate balance after matching or weighting on them. The best practices for assessing balance are described in the main vignette for the cobalt R package, which was designed to assess balance. There is a broad literature on best practices for balance assessment. I recommend reading Ho, Imai, King, and Stuart (2007) for information on how propensity scores should be assessed and Austin (2009) for modern good practices in balance assessment, in addition to the cobalt vignette.
A: Estimation is never correct, but it can be precise. Nothing along the lines of what you're asking for is specially developed for propensity scores; but when you view the problem as one of prediction, there's probably too many methods to count. Asking for test statistics is silly because you don't have a hypothesis. As far as plots, however, you can consider ROC curves, simple boxplots, or diagnostic plots related to the logistic regression model used to obtain the scores.
A: You can easily check with standardized differences or similar statistics whether your propensity score, when used appropriately (eg with matching or inverse weighting), helps you in reducing the apparent bias/confounding due to measured confounders (eg https://atm.amegroups.com/article/view/22865/22385).
However, you may be fooled if a given confounder has been inappropriately labelled or measured (for instance think about the difficulty inherent in labelling sex/gender or race/ethnicity, eg https://ajph.aphapublications.org/doi/10.2105/AJPH.92.9.1471).
Most importantly, no propensity score can adjust for an unmeasured confounder (despite occasional claims for the opposite, eg https://www.jclinepi.com/article/S0895-4356(16)30334-1/fulltext).
In summary, as stated by AdamO and Noah, you may be precise (spuriously?) in your estimation, but accuracy is elusive, with randomized trials being the only really valid approach for hypothesis testing and effect size estimation of interventions.
