Hi 
Both programs add their own extraneous code and interpret such code
differently.
MS Office 2007 is particularly dodgy. It's default format is one of the
OOXML "transitional" formats and the whole program includes some legacy
binary blobs that even MS don't appear to understand.
On MS Office 2007 it might be best to;
1. Use the Golden Globe thing at the top left where all other versions of
MS Office (i'm not sure about 365 or 2016) have a menu called "File"
2. Go down to something like "Save As ..." or "Change Format" or
"Different Format" or whatever they call it.
3. Select the older MS format from 2003, Xp, 2000, 1997. NOT any XML
one!
Really the best bet is to install LibreOffice, OpenOffice or almost any
other Office Suite or word-processing program because almost all of us use
the same format as each other. Then stick to using the ODT format between
the 2 machines.
If you have to then share it with anyone else then THAT is the time to try;
File - "Save As ..."
into a Microsoft format but even then it is better to use the older
Microsoft format otherwise you have no clue how it will look in whichever
version of MS Office is being used by whoever opens it.
Sadly trying the other way around, to install MS Office 2007 on both
machines, may well give you the same problem as you have at the moment.
While installing MS Office 2010 i noticed an MS disclaimer saying the
documents produced by the same program on different OSes might result in
slight differences to the format of the document!
Many people who share files in MS formats have learned to send files as
Pdfs to try to ensure the layout look almost the same on everyone's
machine. Annoyingly when "Saving As ... Pdf" or "Exporting to Pdf" in MS
Office it uses a lossy compression which makes the document a bit blurry
and it is extremely difficult to change that. Also i think MS Office 2007
needs an add-on in order to save as Pdf at all.
With LibreOffice it's MUCH easier to create Pdfs. The "Save As ... Pdf"
dialogue includes many options, such as changing from the lossy (Jpg)
compression to a lossless one so that the resulting Pdf is not blurry. 
Another case where LibreOffice produces better quality documents with less
fuss. 
With LibreOffice you can even tick the box there to make a "hybrid" Pdf.
This looks like a Pdf to everyone but LibreOffice users can open it and
edit it in a smooth word-processing type of way. It's a lot "heavier"
(larger file-size) but it can be handy! 
Of course most people can't edit PDFs as smoothly as they edit with a
word-processor = so if sending a file as Pdf you may need to send an
editable version of the document too! Odt and Doc are editable formats -
avoid DocX!
The upshot of all this is to ditch the MS Office 2007 and just stick with
LibreOffice or a combination of LibreOffice and almost any other non-MS
Office Suite or word-processing program.
Actually it is worth keeping MS Office 2007 but only so that you can open
some of the files that MS die-hards send you! For the most part
LibreOffice (and the rest) all handle MS formatted files just perfectly
fine but nature of MS formats is that they always seem to create problems
for everyone (incl MS Office users) in one or two files.
Regards from
Tom 