It is not uncommon for Xi and Xj to be a) correlated, but both in a b) non-significant regression model.

If you want my opinion, fit two separate regression models once on self-reported and once on diagnosed anxiety.

Try to do this with Minitab software.

To my opinion it does not make much sense to have self-reported and diagnosed anxiety together in the same model. One would expect that they share a considerable amount of information. So my concern here is more the model containing both predictors, not the fact that you analyse them in separate models.

Another option is possibly to combine them, but this may have its own problems.

But maybe you wanted to see if a professional diagnosis adds predcitive value on top of that what is delivered by self-reports. This does not seem to be the case - but this would need the analysis of the predictive power (PPV, NPV, etc.), also using the model with the two predictors.

These variables will be collinear so interpreting their regression effects in the same model will be tricky. This is because the effect of a predictor is the unique effect partialing out all other predictors. If they are highly correlated there is little unique effect and you have very low statistical power to detect those effects. In your case you also probably aren't interested in their unique effects. Some options (not exhaustive):

1) Fit the model with other predictors and add the two anxiety measures as a block. Interpret the change in R^2 from adding both predictors together. This can be interpreted with the simple correlations and or coefficients from separate models.

2) Use latent variable approach. This would be a structural equation model or similar. Include a latent variable for anxiety as a predictor and the two measures of the latent variable and the correlation between them. This will be quite a lot of extra work if you are unfamiliar with latent variable approaches.

3) Reparameterize your two anxiety predictors. Assuming they are on the same scale enter the average of the two measures and their difference. (If not on the same scale rescale them using POMP or z scores first). This gives you an anxiety predictor and a potentially interesting effect of the the difference in diagnosed and self-rated. This is arguably a hack to get something like the latent variable approach. If they are highly correlated the average and the latent variable approach are very unlikely to produce markedly different results (indeed as the correlation approaches 1 the results will converge).

P.S. if they variables are correlated with r = .695 then you have collinearity. A lot of it. If your text book tells you otherwise get a new text book! No multicollinearity means that the correlations between predictors are precisely zero. Any correlation other than zero implies a degree of collinearity. This will impact the statistical power to detect unique effects (maybe only a little if the correlations are low). r = .695 implies just these two predictors lower the efficiency/statistical power quite a bit. .695^2 = 0.483 so they share 48% of variance. This means any test of their unique effects loses almost 50% of the effective sample size. Unless you have a huge sample that's probably a major reduction in statistical power. Collinearity with predictors may reduce this further.