How are you exporting the file? What format and what settings?
I will go ahead and assume PDF, and an older version that includes flattening. Flattened PDFs do not contain transparency, so anyplace transparency is used has to be saved in a way that blends the colours, images, objects, and text to retain the same appearance. This usually requires text to be converted to outlines.
Text in Opentype or Truetype formats includes hinting, which optimises the appearance of text at low resolutions to make it more legible and more closely resemble how it will appear at high resolution. Paths, which is what outlined text is, do not benefit from hinting, so usually look a bit bolder.
Imagine drawing a circle on graph paper, then filling in each grid square within the circle. Some grid lines are entirely outside the circle, so those are white. Some are entirely within the circle, so those are black. But any square with the path of the circle on it must either be white or black. That's the decision your PDF viewer is making when it converts the paths of the text to pixels on your monitor. If the grid squares are very small, the bolding appearance is minimised. The RIP in your printer (assuming the file will be printed on a high resolution press) is very high resolution. The lack of hinting on some text is almost imperceptible.
There are ways to minimise the bolding effect by minimizing where text is outlines when flattened. You can save in a PDF format that does not flatten. You can put the text on a layer above any transparency, so the text is unaffected by flattening. You can use Acrobat to convert all text to outlines, which will remove hinting from all text so there is no contrast between flattened and unflattened text.
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